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Full text of "Higher geometry; an introduction to advanced methods in analytic geometry"

MATH- 

STAT. 

LIBRARY 



HIGHER GEOMETRY 



AN INTRODUCTION TO ADVANCED METHODS 
IN ANALYTIC GEOMETRY; ;, . .-.: 



BY 



FREDERICK S. WOODS 

PROFESSOR OF MATHEMATICS IN THE MASSACHUSETTS 
INSTITUTE OF TECHNOLOGY 




GINN AND COMPANY 

BOSTON NEW YORK CHICAGO LONDON 
ATLANTA DALLAS COLUMBUS SAN FRANCISCO 



COPYRIGHT, 1922, BY FREDERICK S. WOODS 
ALL RIGHTS RESERVED 

322.10 




-G1NN AND COMPANY- PRO- 
PRIETORS BOSTON U.S.A. 



PREFACE 

The present book is the outgrowth of lectures given at various 
times to students of the later undergraduate and earlier graduate 
years. It aims to present some of the general concepts and methods 
which are necessary for advanced work in algebraic geometry (as 
distinguished from differential geometry), but which are not now 
accessible to the student in any one volume, and thus to bridge 
the gap between the usual text in analytic geometry and treatises 
or articles on special topics. 

With this object in view the author has assumed very little 
mathematical preparation on the part of the student beyond that 
acquired in elementary courses in calculus and plane analytic geom 
etry. In addition it has been necessary to assume a slight knowl 
edge of determinants, especially as applied to the solution of linear 
equations, such as may be acquired in a very short course on the sub 
ject. But it has not been assumed that the student has had a course 
in higher algebra, including matrices, linear substitutions, invariants, 
and similar topics, and no effort has been made to include a dis 
cussion of these subjects in the text. This restriction in the tools 
to be used necessitates at times modes of expression and methods 
of proof which are a little cumbersome, but the appeal to a larger 
number of readers seems to justify the occasional lack of elegance. 

In preparing the text one of the greatest problems has consisted 
in determining what matters to exclude. It is obvious that an 
introduction to geometry cannot contain all that is known on any 
subject or even refer briefly to all general topics. The matter of 
selection is necessarily one of individual judgment. One large 
domain of geometry has been definitely excluded from the plan of 
the book ; namely, that of differential geometry. In the field which 
is left the author cannot dare to hope that his choice of material 
will agree exactly with that which would be made by any other 
teacher. He hopes, however, that his choice has been sufficiently 
wise to make the book useful to many besides himself. 

iii 

49H385 



iv PREFACE 

The plan of the book calls for a study of different coordinate 
systems, based upon various geometric elements and classified 
according to the number of dimensions involved. This leads natu 
rally to a final discussion of ^-dimensional geometry in an abstract 
sense, of which the particular geometries studied earlier form con 
crete illustrations. As each system of coordinates is introduced, the 
meaning of the linear and the quadratic equations is studied. The 
student is thus primarily drilled in the interpretation of equations, 
but acquires at the same time a knowledge of useful geometric facts. 
The principle of duality is constantly in view, and the nature of 
imaginary elements and the conventional character of the locus at 
infinity, dependent upon the type of coordinates used, are carefully 
explained. 

Numerous exercises for the student have been introduced. In 
some cases these carry a little farther the discussion of the text, 
but care has been taken to keep their difficulty within the range 
of the student s ability. FREDERICK S. WOODS 



CONTENTS 

PART I. GENERAL CONCEPTS AND ONE-DIMENSIONAL 

GEOMETRY 

CHAPTER I. GENERAL CONCEPTS 

SECTION PAGE 

1. Coordinates 1 

2. The principle of duality 2 

3. The use of imaginaries 2 

4. Infinity 3 

5. Transformations 4 

6. Groups 6 

CHAPTER II. RANGES AND PENCILS 

7. Cartesian coordinate of a point on a line 8 

8. Projective coordinate of a point on a line 8 

9. Change of coordinates i 9 

10. Coordinate of a line of a pencil 11 

11. Coordinate of a plane of a pencil 12 

CHAPTER III. PROJECTIVITY 

12. The linear transformation 13 

13. The cross ratio 16 

14. Harmonic sets 18 

15. Projection 20 

16. Perspective figures 21 

17. Other one-dimensional extents 23 



PART II. TWO-DIMENSIONAL GEOMETRY 
CHAPTER IV. POINT AND LINE COORDINATES IN A PLANE 

18. Homogeneous Cartesian point coordinates 27 

19. The straight line 27 

20. The circle points at infinity 30 

21. The conic 32 

22. Trilinear point coordinates 34 

23. Points on a line 35 

v 



vi CONTENTS 

SECTION PAGE 

24. The linear equation in point coordinates 36 

25. Lines of a pencil 37 

26. Line coordinates in a plane 38 

27. Pencil of lines and the linear equation in line coordinates ... 39 

28. Dualistic relations 40 

29. Change of coordinates 41 

30. Certain straight-line configurations 44 

31. Curves in point coordinates 50 

32. Curves in line coordinates 53 



CHAPTER V. CURVES OF SECOND ORDER AND SECOND CLASS 

33. Singular points of a curve of second order 58 

34. Poles and polars with respect to a curve of second order .... 59 

35. Classification of curves of second order 65 

36. Singular lines of a curve of second class 67 

37. Classification of curves of second class 68 

38. Poles and polars with respect to a curve of second class .... 70 

39. Projective properties of conies 72 

CHAPTER VI. LINEAR TRANSFORMATIONS 

40. Collineations 78 

41. Types of nonsingular collineations 83 

42. Correlations 88 

43. Pairs of conies 95 

44. The projective group 100 

45. The metrical group 101 

46. Angle and the circle points at infinity 105 

CHAPTER VII. PROJECTIVE MEASUREMENT 

47. General principles 107 

48. The hyperbolic case ; 110 

49. The elliptic case 115 

50. The parabolic case 117 

CHAPTER VIII. CONTACT TRANSFORMATIONS IN THE PLANE 

51. Point-point transformations 120 

52. Quadric inversion 121 

53. Inversion 124 

54. Point-curve transformations 127 

55. The pedal transformation 131 

56. The line element . 133 



CONTENTS vii 
CHAPTER IX. TETRACYCLICAL COORDINATES 

SECTION PAGE 

57. Special tetracyclical coordinates 138 

58. Distance between two points , .. .. 139 

59. The circle 140 

60. Relation between tetracyclical and Cartesian coordinates .... 142 

61. Orthogonal circles 144 

62. Pencils of circles 146 

63. The general tetracyclical coordinates ; 150 

64. Orthogonal coordinates . v . . . 153 

65. The linear transformation ; . . . 154 

66. The metrical transformation ........ 155 

67. Inversion . 156 

68. The linear group 159 

69. Duals of tetracyclical coordinates 161 

CHAPTER X. A SPECIAL SYSTEM OF COORDINATES 

70. The coordinate system 164 

71. The straight line and the equilateral hyperbola 166 

72. The bilinear equation : . . . 167 

73. The bilinear transformation . 169 



PAET III. THREE-DIMENSIONAL GEOMETRY 
CHAPTER XI. CIRCLE COORDINATES 

74. Elementary circle co-ordinates . 171 

75. The quadratic circle complex 173 

76. Higher circle coordinates 177 

CHAPTER XII. POINT AND PLANE COORDINATES [ 

77. Cartesian point coordinates 180 

78. Distance 181 

79. The straight line 182 

80. The plane 185 

81. Direction and angle ....... 188 

82. Quadriplanar point coordinates 193 

83. Straight line and plane 194 

84. Plane coordinates 197 

85. One-dimensional extents of points 200 

86. Locus of. an equation in point coordinates 205 

87. One-dimensional extents of planes .. 210 

88. Locus of an equation in plane coordinates . . ** 

89. Change of coordinates . ..... . -^ 



viii CONTENTS 

CHAPTEE XIII. SURFACES OF SECOND ORDER AND OF 
SECOND CLASS 

SECTION PAGE 

90. Surfaces of second order 220 

91. Singular points 221 

92. Poles and polars 222 

93. Classification of surfaces of second order 224 

94. Surfaces of second order in Cartesian coordinates 227 

95. Surfaces of second order referred to rectangular axes 229 

96. Rulings on surfaces of second order 232 

97. Surfaces of second class 235 

98. Poles and polars 238 

99. Classification of surfaces of the second class . . 238 



CHAPTER XIV. TRANSFORMATIONS 

100. Collineations 240 

101. Types of nonsingular collineations 241 

102. Correlations 246 

103. The projective and the metrical groups 249 

104. Projective geometry on a quadric surface 250 

105. Projective measurement 253 

106. Clifford parallels 255 

107. Contact transformations 258 

108. Point-point transformations * 260 

109. Point-surface transformations 262 

110. Point-curve transformations . . 263 



CHAPTER XV. THE SPHERE IN CARTESIAN COORDINATES 

111. Pencils of spheres 266 

112. Bundles of spheres 268 

113. Complexes of spheres 269 

114. Inversion 270 

115. Dupin s cyclide 274 

116. Cyclides 279 



CHAPTER XVI. PENTASPHERICAL COORDINATES 

117. Specialized coordinates 282 

118. The sphere 284 

119. Angle between spheres 286 

120. The power of a point with respect to a sphere 287 

121. General orthogonal coordinates 288 



CONTENTS ix 

SECTION PAGE 

122. The linear transformation 291 

123. Relation between pentaspherical and Cartesian coordinates . . 293 

124. Pencils, bundles, and complexes of spheres 293 

125. Tangent circles and spheres 295 

126. Cyclides in pentaspherical coordinates 297 



PART IV. GEOMETRY OF FOUR AND HIGHER 
DIMENSIONS 

CHAPTER XVII. LINE COORDINATES IN THREE- 
DIMENSIONAL SPACE 

127. The Plticker coordinates ". 301 

128. Dualistic definition 303 

129. Intersecting lines 304 

130. General line coordinates 305 

131. Pencils and bundles of lines 306 

132. Complexes, congruences, series 308 

133. The linear line complex 310 

134. Conjugate lines 314 

135. Complexes in point coordinates 316 

136. Complexes in Cartesian coordinates 317 

137. The bilinear equation in point coordinates 321 

138. The linear line congruence 322 

139. The cylindroid 323 

140. The linear line series 324 

141. The quadratic line complex 328 

142. Singular surface of the quadratic complex 331 

143. Pliicker s complex surfaces 334 

144. The (2, 2) congruence 335 

145. Line congruences in general 336 

CHAPTER XVIII. SPHERE COORDINATES 

146. Elementary sphere coordinates 341 

147. Higher sphere coordinates 343 

148. Angle between spheres 344 

149. The linear complex of oriented spheres 346 

150. Linear congruence of oriented spheres 348 

151. Linear series of oriented spheres 349 

152. Pencils and bundles of tangent spheres 350 

153. Quadratic complex of oriented spheres 353 

154. Duality of line and sphere geometry 357 



x CONTENTS 

CHAPTER XIX. FOUR-DIMENSIONAL POINT COORDINATES 

SECTION PAGE 

155. Definitions 362 

156. Intersections 365 

157. Euclidean space of four dimensions 368 

158. Parallelism 370 

159. Perpendicularity 373 

160. Minimum lines, planes, and hyperplanes 378 

161. Hypersurfaces of second order 382 

162. Duality between line geometry in three dimensions and point 

geometry in four dimensions 384 

CHAPTER XX. GEOMETRY OF N DIMENSIONS 

163. Projective space 388 

164. Intersection of linear spaces 390 

165. The quadratic hypersurface 392 

166. Intersection of a quadric by hyperplanes 396 

167. Linear spaces on a quadric 401 

168. Stereographic projection of a quadric in S n upon S n _ l . . . . 407 

169. Application to line geometry 410 

170. Metrical space of n dimensions 413 

171. Minimum projection of S n upon S n _ 1 419 

INDEX 421 



HIGHER GEOMETRY 

PAET I. GENERAL CONCEPTS AND 
ONE-DIMENSIONAL GEOMETRY- 

11 > 

CHAPTER I 

GENERAL CONCEPTS 

1. Coordinates. A set of n variables, the values of which fix a 
geometric object, are called the coordinates of the object. The ana 
lytic geometry which is developed by the use of these coordinates 
has as its element the object fixed by the coordinates. The reader 
is familiar with the use of coordinates to fix a point either in the 
plane or in space. The point is the element of elementary ana 
lytic geometry, and all figures are studied as made up of points. 
There is, however, no theoretical objection to using any geometric 
figure as the element of a geometry. In the following pages we 
shall discuss, among other possibilities, the use of the straight line, 
the plane, the circle, and the sphere. / 

The dimensions of a system of geometry are determined by the 
number of the coordinates necessary to fix the element. Thus 
the geometry in which the element is either the point in the plane 
or the straight line in the plane is two-dimensional ; the geometry 
in which the element is the point in space, the circle in the plane, 
or the plane in space is three-dimensional ; the geometry in which 
the element is the straight line or the sphere in space is four- 
dimensional. 

Since each coordinate may take an infinite number of values, 
the fact that a geometry has n dimensions is often indicated by 
saying that the totality of elements form an GO" extent. Thus the 
points in space form an co 3 extent, while the straight lines in 
space form an cc 4 extent. If in an oo n extent the coordinates of an 
element are connected by k independent conditions, the elements 

1 



2 ONE-DIMENSIONAL GEOMETEY 

satisfying the conditions form an <x> n ~ k extent lying in the oo w 
extent. Thus a single equation between the coordinates of a point 
in space defines an co 2 extent (a surface) lying in an oo 3 extent 
(space), and two equations between the coordinates of a point in 
space define an oo 1 extent (a curve). 

2. The principle of duality. When the element has been selected 
and its coordinates determined, the development of the geometry 
-consists m" studying the meaning of equations and relations con- 

in^ the coord mates. There are therefore two distinct parts to 
geometry, the analytic work and the geometric interpreta 
tion. Two systems of geometry depending upon different elements 
with the same number of coordinates will have the same analytic 
expression and will differ only in the interpretation of the analy 
sis. In such a case it is often sufficient to know the meaning of 
the coordinates and the interpretation of a few fundamental rela 
tions in each system in order to find for a theorem in one geom 
etry a corresponding theorem in the other. Two systems which 
have such a relation to each other are said to be dualistic, or to 
correspond to each other by the principle of duality. 

It is obviously inconvenient to give examples of this principle 
at this time, but the reader will find numerous examples in the 
pages of this book. 

3. The use of imaginaries. Between the coordinates of a geo 
metric element and the element, itself there fails to be perfect equiv 
alence unless the concept of an imaginary element is introduced. 
Consider, for example, the usual Cartesian coordinates (z, y) of a 
point in a plane. If we understand by a " real point " one which has 
a position on the plane which may be represented by a pencil dot, 
then to any real pair of values of x and y corresponds a real point, 
and conversely. It is highly inconvenient, however, to limit our 
selves in the analytic work to real values of the variables. We 
accordingly introduce the convention of an " imaginary point " by 
saying that a pair of values of x and y of which one or both is a 
complex quantity defines such a point. In this sense a " point " 
is nothing more than a concise expression for " a value pair (#, ^)." 
From this standpoint many propositions of analytic geometry 
are partly theorems and partly definitions. For example, take the 
proposition that any equation of the first degree represents a straight 



GENERAL CONCEPTS 3 

line. This is a theorem as far as real points and real lines are 
concerned, but it is a definition for imaginary points satisfying an 
equation with real coefficients and for all points satisfying an equa 
tion with complex coefficients. The definition in question is that 
a straight line is the totality of all value pairs (x, y) which satisfy 
any linear equation. 

Any proposition proved for real figures may be extended to imag 
inary figures provided that t;he proof is purely an analytic one 
which is independent of the reality of the quantities involved. 
One cannot, however, extend theorems which are not analytic in 
their nature. For example, it is proved for a real triangle that the 
length of any side is less than the sum of the lengths of the 
other two sides. The length of the side connecting the vertices 
(x^yj and (o? 2 , ?/ 2 ) is V(^~ a^ +C^- # 2 ) 2 . We may extend 
this definition of length to imaginary points, but the theorem con 
cerning the sides of a triangle cannot be proved analytically and 
is not true for imaginaries, as may be seen by testing it for the 
triangle whose vertices are (0, 0), (i, 1), and (i, 1). 

Similar considerations to those we have just stated for a point 
in a plane apply to any element. It is usual to have a real element 
represented by real coordinates, but sometimes it is found con 
venient to represent a real element by complex coordinates. In 
either case there will be found in the analysis certain combinations 
of coordinates which cannot represent real elements. In all cases 
the geometry is extended by the convention that such coordinates 
represent imaginary elements. 

4. Infinity. Infinity may occur in a system of geometry in two 
ways : first, the value of one or more of the coordinates may increase 
without limit, or secondly, the element which we suppose lying 
within the range of action of our physical senses may be so displaced 
that its distance from its original position increases without limit. 

Infinity in the first sense may be avoided by writing the coor 
dinates in the form of ratios, for a ratio increases without limit when 
its denominator approaches zero. Coordinates thus written are called 
homogeneous coordinates, because equations written in them become 
homogeneous. They are of constant use in this book. 

The treatment of infinity in the second sense is not so simple, 
but proceeds as follows: As an element of the geometry recedes 



4 ONE-DIMENSIONAL GEOMETRY 

indefinitely from its original position, its coordinates usually 
approach certain limiting values, which are said by definition to 
represent an " element at infinity." The coordinates of all ele 
ments at infinity usually satisfy a certain equation, which is said 
to represent the " locus at infinity." The nature of this locus 
depends upon the coordinate system. Thus, in the plane, by the 
use of one system of coordinates all " points at infinity " are said 
to lie on a " straight line at infinity " ; by another system of coor 
dinates the plane is said to have " a single real point at infinity " ; 
by still another system of coordinates the plane is said to have 
" two lines at infinity." These various statements are not contra 
dictory, since they are not intended to express any fact about the 
physical properties of the plane. They are simply conventions to 
express the way in which the coordinate system may be applied 
to infinitely remote elements. There is no more difficulty in pass 
ing from one convention to another than there is in passing 
from one coordinate system to another. The convention as to 
elements at infinity stands on the same basis as the convention as 
to imaginary elements. 

5. Transformations. A transformation is an operation by which 
each element of a geometry is replaced by another element. The 
new element may be of the same kind as the original element or 
of a different kind. For example, a rotation of a plane about a 
fixed point is a transformation of points into points ; on the other 
hand, a transformation may be made in the plane by which each 
point of the plane is replaced by its polar line with respect to a 
fixed conic. We shall consider in this book mainly analytic trans 
formations^ that is, those in which the coordinates of the trans 
formed element are analytic functions of those of the original 
element. 

A transformation may be conveniently expressed by a single 
symbol, such as T. If we wish to express the fact that an element, 
or a configuration of elements, a, has been transformed into another 
element or configuration &, we write 

T() = 6. (1) 

Suppose now, having carried out the transformation T, we carry 
out on the transformed elements another transformation S. The 



GENERAL CONCEPTS 5 

result is a single transformation G 9 and we write 

G = ST, (2) 

where G is called the product of S and T. 

Similarly, the carrying out in succession of the transformation 
T, then S, and then R, is the product RST. This symbol is to be 
interpreted as meaning that the transformations are to be carried 
out in order from right to left. This is important, as the product 
of transformations is not necessarily commutative. For example, let 
T be the moving of a point through a fixed distance in a fixed 
direction and S the replacing of a point by its symmetrical point 
with respect to a fixed plane. It is evident in this case that 

ST^TS. (3) 

A product of transformations is, however, associative. To prove this, 
let R, S, and T be three transformations. We wish to show that 

(fiS)T=(ST) = fiST. (4) 

In the sense of formula (1) let 



Then (72S) T(a) = S(b) *=R(c) = d. 

On the other hand, ST(a) = S(b)= c, 
so that R(ST) (a) = JK (<?) = & 

This establishes the theorem. 

If T represents an operation, T~ l shall represent the inverse 
operation ; that is, if T transforms any element a into an element 
b, T~* shall transform every element b back into the original a. 
The product then of T and T~ l in any order leaves all elements 
unchanged. It is natural to call an operation which leaves all ele 
ments unchanged an identical transformation and to indicate it by 
the symbol 1. We have then the equation 

TT- l = T~ l T = l. (5) 

If S and T are two transformations, the operation 

TST~ l = S (6) 

is called the transform of S by T. 

If S[ and $2 are the transforms of S 1 and S. 2 respectivelj^fneii 
S[S^ is the transform of S^ For ^ 

. T~\ 



6 ONE-DIMENSIONAL GEOMETRY 

EXERCISES 

1. State which of the following pairs of operations are commutative : 

(a) a translation and a rotation about a fixed point ; ^ 

() two rotations ; ^" 

(c) two translations ; 

(d) a rotation and a reflection on a line. "^ " 

2. If S is a transformation such that S 2 = 1, prove that S~ l S, and 
conversely. Give geometric examples of transformations of this type. 

3. Prove that the reciprocal of the product of two transformations is 
the product of the reciprocals of the transformations in inverse order ; 
that is, prove that (RST)~ l = T- l S~ l R-\ 

4. If S is a rotation in a plane and T a translation, find the trans 
form of S by T and the transform of T by S. 

5. Prove that the transform of the inverse of S is the inverse of the 
transform of S. 

6. If the product of two transformations is commutative, show that 
each is its own transform by the other. 

6. Groups. A set of transformations form a group if the set contains 
the inverse of every transformation of the set and if the product of any 
two transformations of the set^is also a transformation of the set/^fy.& 

In general the definition of a group of operations involves also 
the conditions that the operations shall be associative and that the 
identical transformation shall be defined. These latter conditions 
being always true for geometrical transformations need not be 
specified in our definition nor explicitly looked for in determining 
whether or not a given set of transformations form a group. 

As an example of a group consider the operations consisting of 
rotating the points in space around a fixed axis through any angle 

equal to any multiple of - - Another example consists of all 

possible rotations around the same axis. 

A set of operations forming a group and contained in a larger 
group form a subgroup of the larger group. For example, the rota 
tions about a fixed axis through multiples of - form a subgroup 

5 

of all rotations about the same axis. Again, all mechanical motions 
in space form a group. All translations form a subgroup of the 



GENERAL CONCEPTS 7 

group of mechanical motions. All translations in a fixed direction 
form a subgroup of the group of translations and hence a sub- 
subgroup of the group of motions. , 

The importance of the concept of group Jin geometry lies in the 
fact that it furnishes a means of classifying different systems of 
geometry. The element of the geometry having been chosen, any 
group of transformations may be tal^en, and the properties of 
geometric figures may be studied which are unaltered by all trans 
formations of the group. Thus tke ordinary geometry of space 
considers the properties of figure> which are unaltered by the group 
of mechanical movements. f 

Any property or configuration which is unaltered by the opera 
tions of a group is called an invariant of the group. Thus distance 
is an invariant of the group of mechanical motions, and a circle is 
an invariant with respect to the group of rotations in the plane 
of the circle about the center of the circle. 

EXERCISES 

1. If x is the distance of a point P on a straight line from a fixed 
point 0, and P is transformed into a new point P such that x = ax 4- 6, 
prove that the set of transformations formed by giving to a and b all 
possible values form a group. 

2. If (a;, y) are Cartesian coordinates in a plane, and a transformation 
is expressed by the equations 

x = x cos a y sin a, 
y = x sin a -j- y cos a, 

prove that the transformations obtained by giving a all possible values 
form a group. 

3. If (x, y) are Cartesian coordinates in a plane, prove that the 
transformations defined by the equations 

x = x cos a + y sin a, 

y = x sin a y cos a, 
do not form a group. 

4. Name some subgroups of the groups in Exs. 1-2. 

5. Let G be a given group and G l a subgroup. If every transforma 
tion of G l is replaced by its transform by T, where T belongs to G, show 
that the transformations thus found form a subgroup of G. 



\ 

CHAPTER II 

RANGES AND PENCILS 

7. Cartesian coordinate of a point on a line. Consider all points 
which lie on a line LK (Fig. ). These points are called a pencil 
or a range, and the line LK is called the axis or the base of the 
range. Any point P on LK may A o B p 

be fixed most simply by means of L ~ " ~* ^ 

its distance OP from a fixed origin 

0, the distance being reckoned positive or negative according as P 
lies on one side or another of 0. We may accordingly place 

x=OP (1) 

and call x the coordinate * of P. To any point P corresponds one 
and only one real coordinate x, and to any real x corresponds 
one and only one real point P. Complex values of x are said, as 
in 3, to define imaginary points on LK. 

The coordinate may be made homogeneous (4) by using 

the ratio x : t, where - = OP. As P recedes indefinitely from 0, t 
t 

approaches the value 0. Hence, as in 4, we make the convention 
that the line has one point at infinity with the coordinate 1 : 0. 
When the nonhomogeneous x of (1) is used, the point at infinity 
has the coordinate oo. 

The coordinate x we call the Cartesian coordinate of P because 
of its familiar use in Cartesian geometry. 

8. Projective coordinate of a point on a line. On the straight 
line LK (Fig. 1) assume two fixed points of reference A and B 
and two constants k l and & 2 . Then if P is any point on LK we 
may take as the coordinate of P the ratio x l : a? 2 , where 

x^.x^k^AP-.k^BP, (1) 

* The word " coordinate " may be objected to on the ground that it implies the 
existence of at least two quantities which are coordinated in the usual sense. In 
spite of this objection we retain the word to emphasize the fact that we have here 
the simplest case of coordinates in an n-dimensional geometry. 

8 



RANGES AND PENCILS 9 

in which the distances AP and BP are positive or negative accord 
ing as P is on the one side or the other of A or B respectively. 
It is evident that the correspondence between real points on LK 
and real values of the ratio x l : # 2 is one to one. Complex values 
of the ratio define imaginary points on LK (3). 

The Cartesian coordinate of the preceding article may be con 
sidered as a special or limiting case of the kind just given. For if 
in (1) we place Jc 1 = 1, allow the point B to recede to infinity, 
and at the same time allow & 2 to approach zero in such a manner 
that the limit of & 2 BP remains finite, equations (1) give the 
homogeneous Cartesian coordinates of P. 

Considering (1), we see that as P recedes indefinitely from A 
and B the ratio x l : % 2 approaches the limiting ratio Jc 1 : & 2 . Hence 
we say that the line has one point at infinity. 

It is to be noticed that the ratio (which alone is essential) of 
the constants k l and & 2 is determined by the coordinate of any one 
point. Since this ratio is arbitrary the coordinate of any point may 
be assumed arbitrarily after the points of reference are fixed. 

In particular any point may be given the coordinate 1:1. This 
point we shall call the unit point. The coordinate of A is : 1 and 
that of B is 1 : 0. Since the unit point and the points of reference 
are arbitrary, it follows that in setting up the coordinate system any 
three points may be given the coordinates 0:1, 1:0, and 1 : 1 respec 
tively, and the coordinate system is fully determined by these points. 

The coordinate of this section we shall call the projective 
coordinate of P because of its use in projective geometry. 

EXERCISES 

1. Establish a coordinate system on a straight line so that the point B 
is 5 inches to the right of A and the unit point 1 inch to the right of A. 
Where is the coordinate negative ? 

2. Take the point B as in Ex. 1 and the unit point 1 inch to the 
right of B. What are the coordinates of points respectively 1, 2, 3, 
4 inches to. the right of A and 1, 2, 3 inches to the left of A ? 

9. Change of coordinates. The most general change from one sys 
tem of projective coordinates to another may be made by changing 
the points of reference and the unit point, the latter change being 
equivalent to changing the ratio of the constants ^ and & 2 . Let 



10 ONE-DIMENSIONAL GEOMETRY 

x l : x 2 be the coordinate of any point P (Fig. 2) referred to the 
points of reference A and B, with certain constants k l and & 2 , and 
let x( : x be the coordinate of the same 

A R ~P 

point referred to the points of reference J - J - -g - - - - 
A and 5 , with constants k( and & 2 . 

-b IG. 2 

Assume any point and let OA= a, 

OA = a 1 , OB = b, OB = V, and OP = t. Then from (1), 8, we have 
x, : x,=k^t - a) : h(t - ft), a{ : a =^( - ) : V 2 (t-V). (1) 
The. elimination of from these equations gives relations of the 
form = ^ 



which are the required formulas for he change of coordinates. 

The ratio of the coefficients a^ a , yS t , and /3 2 will be determined 
if we know three values of x l : x 2 which correspond to three values 
of x( : #2, in particular to the three values 0:1, 1:0, 1:1. For 
when x( : x 2 = : 1 we have x^ : x 2 = a t2 : /3 2 ; when x[:x r z = l:Q we have 
x t : x 2 = a l : f$ l ; and when x\ : x 2 =1 : 1 we have x l : x 2 = a 1 + a. 2 : ft l 4- /3 2 . 

It is obvio.us from the foregoing that if the reference points A 
and B are distinct, the coefficients in (2) must satisfy the condition 
afi^ #2/3^ 0, which is also necessary in order that the ratio x l : x^ 
in equations (2) should contain x[ : x 2 . 

Equations (2) may be placed in a form which is of frequent use. 
Let us place x[:x z = X, a 1= = ^, 1= z 2 , a 2 = y^ /8 2 = y# where y l : y^ 
and z 1 : z z are the coordinates of the two points corresponding to 
X = and X = oo respectively. Then equations (2) become 

^= yi +H, 

p^=%+H- 

Hence, if y 1 : y 2 and z 1 : z 2 are the coordinates of any two points on 
a straight line, the coordinate of any other point may be written 

2/i+ x v# 2 +H- 

EXERCISES 

1. Find the formulas for the change from the coordinate in Ex. 1, 8, 
to that in Ex. 2. 

2. Find the formulas for a change from the coordinate in Ex. 1, 8, 
to one in which the reference points are respectively 2 and 6 inches 
from A and the unit point 4 units from A. 

3. Prove that all changes of coordinates form a group. 




RANGES AND PENCILS 11 

10. Coordinate of a line of a pencil. Consider all straight lines 
which lie in a plane and pass through the same point (Fig. 3). 
Such lines form a pencil, the common point being called the vertex 
of the pencil. 

Let OM be a fixed line in the pencil, OP any line, and 6 the angle 
MOP. Then it would be possible to take 6 as the coordinate of OP, 
but in that case the line OP would 
have an infinite number of coordi 
nates differing by multiples of 2 TT. 
We may make the relation between 
a line and its coordinate one to one 
by taking as the coordinate a quan 
tity x defined by the equation 

x = k tan 0, (1) 

where k is an arbitrary constant. 

Then x = is the line OM, x = oo is FIG 

the line at right angles to OM, and 

any positive or negative real value of x corresponds to one and 

only one real line of the pencil, and conversely. Imaginary values 

of x define imaginary lines of the pencil as in 3. 

A more general coordinate may be obtained by using two fixed 
lines of reference OA and OB and defining the ratio x l : x 2 by the 
equation ^ . x ^ = ^ s i n AOP:\ sin BOP. (2) 

Equation (2) reduces to equation (1) when the angle AOB is a 
right angle, OA coincides with OM, and x l : x 2 = x. 

In general let the angle MO A = a and the angle MOB = ft. Then 
(2) may be written 

Xl : x z = ^ sin (0 - a) : & 2 sin (0 - /3) 

= Ic^x cos a k sin #) : k 2 (x cos ft Jc sin /:?), (3) 

when x is defined by (1). 

Now let x( : x 2 be another coordinate of the lines of the pencil of 
the same form as in equation (2), but referred to lines of reference 
OA and OB and with constants k[ and k f f Then x{ : x 2 is connected 
with x l : x 2 by a bilinear relation of the form 

, 



12 



ONE-DIMENSIONAL GEOMETRY 



This follows from the fact that both x : z. 2 and x( : x 2 are con 
nected with x by a relation of the form (3). 

Since a transformation of coordinates is effected either by change 
of the liffelTbf reference or by change of the constants ^ and & 2 , it 
follows that any transformation of coordinates is expressed by a 
relation of form (4). The coefficients of the transformation are 
determined when the values of x l : x 2 are known which correspond 
to three values of x[ : Xy The proof is as in 9. Also, as in 9, it 



may be shown that if ^ : y z 



z 2 are "the coordinates of any 



two lines of a pencil, the coordinate of any line may be written 



(5) 



11. Coordinate of a plane of a pencil. Consider all planes which 
pass through the same straight line (Fig. 4). Such planes form a 
pencil or sheaf, and the straight line is called the axis of the pencil. 
The coordinate of a plane of the sheaf may be 
obtained by first assuming two planes of refer 
ence a and b and a fixed constant k. Then, if p 
is any plane of the pencil and (a, jt?) means the 
angle between a and p, we may define the coordi 
nate of p as the ratio x 1 : x z given by the equations 




x i : x * = \ sin ( a P) : 



sn 



FIG. 4 



It is obvious that if a plane m be passed per 
pendicular to the axis of the pencil, the planes of 
the pencil cut out a pencil of lines in the plane m. 
The angle between two lines of this pencil is the 
plane angle of the two planes in which the two lines lie. Hence 
the coordinate x l : x 2 defined in (1) is also the coordinate of the 
lines of the pencil in the plane m, in the sense of 10. The results 
of 10 with reference to transformation of coordinates hold, there 
fore, for a pencil of planes. In particular, if y l : # 2 and z l : z 2 are 
the coordinates of any two planes of a sheaf, the coordinate of any 
plane of the pencil may be written 



(2) 



CHAPTER III 

PROJECTIVITY 

12. The linear transformation. We shall now consider the 
substitution 

-- c^,-A*o) a) 



not as a change of coordinates, as in 9, but as denning a trans 
formation in the sense of 5. Then x l : x 2 are to be interpreted as 
the coordinate of an element of a one-dimensional extent and 
x( : x f 2 as the coordinate of the transformed element of the same or 
another one-dimensional extent. If x l : x 2 and x[ : x 2 refer to dif 
ferent extents, the elements need not be of the same kind. For 
example, the transformation (1) may express the transformation 
of points into lines, of points into planes, of lines into planes, and 
so on. 

To study the transformation we shall find it convenient to use 
a nonhomogeneous form obtained by replacing x l : x 2 by X, x( : x 2 
by X , and changing the form of the constants. We have 



Here X and X may be the point, line, or plane coordinates of 
7, 8, 10, 11 or may be the X used in the formulas of 9-11. 
More generally still, X may be any quantity which can be used 
to define an element of any kind, even though not yet employed 
in this text. 

In each case the element with coordinate X is said to be trans 
formed into the element with coordinate X , and the two elements 
X and X are said to correspond. There is one and only one element 
X corresponding to an element X. Conversely, from (2) we obtain 



. 

- 7 X + a 
13 



(3) 



14 ONE-DIMENSIONAL GEOMETRY 

Hence to an element X corresponds one and only one element X. 
In other words, the correspondence between the elements \ and the 
elements X is one to one. 

Any element whose coordinate is unchanged by the trans 
formation is called a fixed element of the transformation. This 
definition has its chief significance when the elements X and X are 
points of the same range, or lines of the same pencil, or planes 
of the same pencil. If, for example, X and X are points of the 
same range, the point X is transformed into the point X , which 
is in general a different point from X, but the fixed points are 
unchanged. 

To find the fixed elements we have to put X = X in (2) or in (3). 

There results 

7 X 2 + (S-a) X-=0. (4) 

Any linear transformation has, accordingly, two fixed elements, which 
may be distinct or coincident. 

If a, yS, 7, and 5 are real numbers, and real coordinates X and X 
correspond to real elements, we may make the following classifica 
tion of the linear transformations : 

(1) (S a) 2 + 4 /ity > 0. The fixed elements are real and distinct. 
The transformation is called hyperbolic. 

(2) (8 o:) 2 + 4 $7 < 0. The fixed elements are imaginary with 
conjugate imaginary coordinates. The transformation is called 
elliptic. 

(3) (8 #) 2 4- 4 fiy = 0. The fixed points are real and coincident. 
The transformation is called parabolic. 

By the transformation (2) an element P with coordinate X is 
transformed into an element Q with the coordinate X . At the 
same time the element Q is transformed into an element R with 
coordinate X". In general, R is distinct from P, for X" is given 
by the equation 

= a\ + ^ (a 2 + /3y*)\ + g/3 + /3S 

2 



In order that X" should always be the same as X it is necessary 
and sufficient that the equation 

X 2 +(S 2 - a 2 ) X - (a+8)= 



PKOJECTIVITY 15 

should be true for all values of X. The coefficients #, & 7, and 8 
must then satisfy the equations 

ay -f 78 = 0, 
S 2 -a 2 =0, (6) 



The second equation gives 8 = a. If we take 8 = a the other 
two equations give 7 = 0, yS= 0, and the transformation (1) reduces 
to the identical transformation X = X . We must therefore take 
S = a, and all three equations (6) are satisfied. 

The transformation then becomes 



(7) 



7 A, CC 

A linear transformation of this type is called involutory. It has 
the property that if repeated once it produces the identical trans 
formation. The correspondence between the elements X and the 
transformed elements X is called an involution. 

EXERCISES 

1. Find !he transformation which transforms 0, 1, co into 1, oo, 0, 

respectively. What are the fixed points of the transformation ? 
" 

2. If x is the Cartesian coordinate of a point on a straight line, 

determine tjfe linear transformation which interchanges the origin and 
the point at infinity. What are the fixed points of the transformation ? 
Do all such transformations form a group ? 

3. If cc is the Cartesian coordinate of a point on a straight line, 
determine the transformation which has only the origin for a fixed 
point and also that which has only the point at infinity for a fixed 
point. Does each of these types of transformation form a group ? 

4. If x is the Cartesian coordinate of a point on a straight line, 
determine a transformation with the fixed points i. Do these form 
a group ? 

5. Show that the general linear transformation may be obtained as 
the product of two transformations of the type X = X, two of the 

type X = X -f b, and one of the type X = - 

A 

6. Show that any transformation with two distinct fixed elements 

, . X a X a 

a and b can be written - - = k - - 
X b X 



16 ONE-DIMENSIONAL GEOMETRY 

7. Show that any transformation with a single fixed element a 

can be written - = - -|- b. 
X a X a 

8. Show that any involutory transformation can be written 

= > where a and b are the fixed elements. 
A o A o 

9. Show that all transformations with the same fixed elements 
form a group. 

10. Consider the set of circles which pass through the same two 
fixed points, and the common diameter of the circles. Show that if P 
and Q are the two points in which any one of the circles meets the 
common diameter, P may be transformed into Q by an involutory 
transformation, the form of which is the same for all points P. Show 
that the transformation is elliptic or hyperbolic according as the two 
fixed points in which the circles intersect are real or imaginary. 

11. Show, conversely to Ex. 10, that any involutory transformation 
may be geometrically constructed as there described. 

13. The cross ratio. The linear transformation contains three 
constants ; namely, the ratios of the four coefficients #, /3, 7, and 8. 
These constants can be so determined that any three arbitrarily 
assumed values of X can be made to correspond to any, three arbi 
trarily assumed values of X . In other words, 

/. By a linear transformation any three elements can be transformed 
into any other three elements, and these three pairs of corresponding 
elements are sufficient to fix the transformation. 

To write the transformation in terms of the coordinates of three 
pairs of corresponding elements, we write first 



v-x; vA 

which is obviously a transformation by which \ is transformed 
into \[, and X 2 into Xj. If, in addition, X 3 is to be transformed into 
Xg, a must be determined by the equation 

(2) 



From (1) and (2) we have 



X - Xj Xg - Xg X - Xj Xg 

which is the required transformation. 



PROJECTIVITY 17 

If X 4 and \4 are a fourth pair of corresponding elements, we have, 
from (3), x ;_ x / _ X^-X; = X 4 - X 2 X 3 -X, ] 

Aj A*< ^o A-o A^ Af- A*o 2 

or, with a slight rearrangement, 




Xj X 4 X. 2 X 3 
The quantity ^ 

A l~ \ 

is called the cross ratio, or the anharmonic ratio, of the four ele 
ments X x , X 2 , X g , X 4 , and is denoted by the symbol (X^, X 3 X 4 ). 
Equation (4) establishes the theorem : 

//. The cross ratio of four elements is unaltered by any linear 
transformation. 

The cross ratio is accordingly independent of the coordinate 
system used in defining the elements. 

The cross ratio depends not only on the four elements involved 
but also on the order in which they are taken. Now four things 
may be taken in twenty-four different orders, but there result only 
six distinct cross ratios. In fact, it is easy to show, by writing all 
possible cross ratios, that the six distinct ones are 

1 1 r-1 r 

r, -^ \ r, - - - - -i 
r ji r r r 1 

where r is any one of them. 

In naming the cross ratio of four elements it is therefore neces 
sary to indicate the order in which the elements are to be taken. 
We have adopted the convention that if .ZJ, P%, P%, and P are four 
elements with the coordinates \, X 2 , X 3 , and X 4 respectively, the 
cross ratio indicated by the symbol (^^, P^J^) shall be given by 
the relation 

.. (6) 



If, then, we denote (^JZJ, P Z I) by r, it is evident that 
v JJJ) = -, (JJJJ, 2JJJ) = 1 - r, 



V l 4 ay t 



18 ONE-DIMENSIONAL GEOMETRY 

A special form which the cross ratio takes for certain coordinates 
is of importance and is given in the following theorem : 

///. If the dements P and Q have the coordinates y^ : y^ and z l : z 2 re 
spectively, and the elements R and Shave the coordinates y l + ^ 1 : y^ + ^^ 2 
and y^ + pz^ : y^ + /z2 2 respectively, then 

(PC, RS) = (RS,PQ) = -- 

To prove this take \ = for the element P, X 2 = oo for the 
element Q, X g =X for the element R, and \ 4 = /* for the element 
S, and substitute in (6). 

If \ is the Cartesian coordinate of a point on a straight line, 
then \-\=P 3 P 19 \ 1 -\=P 4 P 11 \-\ 3 =P 3 P 2 , \-\=P^P 21 and 



The cross ratio is accordingly found by finding the ratio of the 
segments into which the line P^ is divided by PI and the ratio of 
the segments into which P^ is divided by P, and forming the ratio 
of these ratios. 

14. Harmonic sets. If a cross ratio is equal to 1, it is called 
a harmonic ratio. If P v P z , P & , and P are four elements such that 



the four elements form a harmonic set, and the points P^ and P 2 
are said to be harmonic conjugates to P z and P. 

From III, 13, it follows that the points y^+ \z^:y 2 + \z z and 
y\ ~ ^i : #2 ~~ Mz are harmonic conjugates to y^ : y z and z l : Z 2 . 

From (7), 13, it follows that if four points on a straight 
line form a harmonic set, then 






This shows that the two points in a harmonic set divide the dis 
tance between their harmonic conjugates internally and externally 
in the same ratio. 



. 



PROJECTIVITY 19 

EXERCISES 

1. Show that the cross ratio of any point, the transformed point, 
and the two fixed points of any elliptic or hyperbolic transformation 
is constant. This is sometimes called the characteristic cross ratio of 
the transformation. What happens to the characteristic cross ratio as 
the two fixed points approach coincidence ? 

2. Show that by any involutory transformation any element is 
transformed into its harmonic conjugate with respect to the two fixed 
elements. 

3. If \ v A 2 , X 3 , X 4 form a harmonic set, prove that 

1 1 



y/ 



In general, prove that if (X^, X 3 X 4 ) = A:, 

1-k 1 k 

A 2 ~~ A l A 4 ~~ A l A 3 ~~ A l 

4. Write the transformation by which each point on a line is trans 
formed into its harmonic conjugate with respect to the points X = a, 
X = a. What are the fixed points of the transformation ? 

5. Prove that an involution of lines of a pencil contains one and 
only one pair of perpendicular lines (that is, one case in which a line 
is perpendicular to its transformed line) unless all pairs of lines are 
perpendicular. When does the latter case occur ? 

6. Let x l : x 2 be the coordinate of a point on a line and consider the 
point pair defined by the equation 

- a 99 ar| = 0. 



Show that the equation may be reduced to one of three types by a 
real transformation of coordinates and give the analytic condition for 
each type. 

7. Let A and B be two distinct points defined by the equation of 
Ex. 6, and P (y l : 7/ 2 ) and Q (^ : 2 ) and R (w l : w 2 ) any three points. If 
the protective distance between two points is defined by the equation 

k 
D(PQ) = - log (PQ, AB), show that D(PQ) + D(QR) = D(PR). 

Consider two cases : 

1. A and B real. Take k real. Then any two points between A and 
B have a real distance apart. A and B are at an infinite distance from 
any other point. Any point not between A and B is at an imaginary 
distance from any point between A and B. 



20 ONE-DIMENSIONAL GEOMETRY 

2. A and B conjugate imaginary. Take k pure imaginary. Any two 
real points are at a real finite distance apart. The total length of the 
line is finite. 

8. Consider the point pair defined by the equation 

a \\ x i 4- 2 a^x.2 + a 22 a;| = 0. 
Then, if y l : y >2 is any given point, the equation 

^1 4- Ol2?/l 4- 



defines a point which is called the polar point of ?/ with respect to 
the point pair. Assuming a n a 22 a^ ^= 0, show that to any point cor 
responds a definite polar point and that any point is the polar point 
of a definite point y. Show that a point and its polar are harmonic 
conjugates with respect to the point pair. What happens to these 
theorems if a n a 22 a? 2 = ? 

15. Projection. Two one-dimensional extents are said to be in 
projection if the elements of the two extents are brought into 
correspondence by means of a linear relation, 



between their coordinates. The correspondence is called a projec- 
tivity. If the correspondence is involutory, the projectivity is an 
involution ( 12). From the definition the following theorems 
may be immediately deduced: 

/. The cross ratio of any four elements of a one-dimensional extent 
is the same as the cross ratio of the four corresponding elements of a 
protective extent. 

II. Two one-dimensional extents may be brought into projection with 
each other in such a way that any three elements of one are made to 
correspond to any three elements of the other. 

III. A projectivity is fully determined by three pairs of corresponding 
elements. 

IV. Two extents ivhich are in projection with the same third extent 
are in projection with each other. 

EXERCISE 

If the points of a circle are connected to any two fixed points of the 
circle, show that the two pencils of lines formed are projective. 



PROJECTIVITY 



21 



16. Perspective figures. A simple case of a projectivity is that 
called a perspectivity, now to be denned. Noting that we have to 
do with pencils of different kinds, 
according as they are made up 
of points, lines, or planes, we 
say that two pencils of different 
kinds are in perspective when 
they are made to correspond in 
such a manner that each element 
of one pencil lies in the corre 
sponding element of the other. 
Two pencils of the same kind FlG 

are in perspective when each is 

in perspective to the same pencil of another kind. The corre 
spondence between perspective figures is called a perspectivity. 

A pencil of points and one of lines are therefore in perspective 
when they lie as in Fig. 5, where the lines a, 5, c, d, etc. correspond 
to the points A, B, C, D, etc. To see that we are justified in calling 
this relation a projectivity, note that 




D \ 



AD 
ED 



OA sinAOD 
OB sin BOD 



Hence, if A and B are taken as fixed points and D as any point, 
the variable X is a coordinate at the same time of the points of the 
pencil of points and of the lines 
of the pencil of lines. Since any 
change of coordinate of either of 
the pencils is expressed by a 
linear relation, the two pencils 
satisfy the definition of projec- 
tive figures. 

Two pencils (ranges) of points 
are in perspective when they are 
perspective to the same pencil 
of lines as in Fig. 6. The straight FlG 6 

lines connecting corresponding 

points of the two ranges then pass through a common point. That 
the relation is a projectivity follows from IV, 15. 




22 



ONE-DIMENSIONAL GEOMETRY 




FIG. 7 



Two pencils of lines are in perspective when they are in per 
spective to the same range of points as in Fig. 7. The points 
of intersection of corresponding 
lines of the two pencils then lie 
on the same straight line. That 
the relation is a projectivity 
follows from IV, 15. 

From these definitions the 
following theorems are easily 
proved : 

/. If four lines of a pencil of 
lines are cut ly any transversal, 
the cross ratio of the four points of 
intersection is independent of the 
position of the transversal and is equal to the cross ratio of the four lines. 

II. If four points of a range are connected with any center, the cross 
ratio of the four connecting lines is independent of the position of the 
center and is equal to the cross ratio of the four points of the range. 

III. If the straight lines connecting three pairs of corresponding points 
of two projective ranges meet in a point, all the lines connecting corre 
sponding points meet in that point, and the ranges are in perspective. 

IV. If the points of intersection of three pairs of corresponding lines 
of two projective pencils lie on a straight line, the points of intersection 
of all pairs of corresponding lines lie on that line, and the pencils are 
in perspective. 

The last two theorems follow from III, 15. 

A pencil of lines is in perspective to a pencil of planes when the 
vertex of the pencil of lines lies in the axis of the pencil of planes 
and each line corresponds to the plane in which it lies. If the plane 
of the pencil of lines is perpendicular to the axis of the pencil of 
planes, the correspondence is a projectivity, since, by 11, the same 
coordinate may be used for each pencil. If the plane of the pencil 
of lines is not perpendicular to the axis of the pencil of planes, the 
pencil of lines is clearly in perspective to another pencil of lines 
with its plane so perpendicular, for in Fig. 7 the two pencils are 
not necessarily in the same plane. Hence the relation here is also 
a projectivity. 



PROJECTIVITY 23 

EXERCISES 

1. Consider any two project! ve pencils of lines not in perspective 
and construct the locus of the intersections of corresponding lines. 
Show that this locus passes through the vertices of the two pencils and 
that it is intersected by an arbitrary line in not more than two points. 

2. Consider any two pencils of points not in perspective and con 
struct the lines joining corresponding points. These lines envelop a 
curve. Show that not more than two of these lines pass through any 
arbitrary point and that the two bases of the pencils belong to these lines. 

3. Consider the locus of the lines of intersection of corresponding 
planes of two pencils of planes not in perspective. Show that this locus 
contains the two axes of the pencils and that it is cut by any arbitrary 
plane in a curve such as is defined in Ex. 1. 

4. Show that if the line connecting the vertices of two project! ve 
pencils of lines is self-corresponding (that is, considered as belonging 
to one pencil it corresponds to itself considered as belonging to the 
other pencil) the pencils are in perspective. 

5. Show that if the point of intersection of the bases of two projective 
ranges is self-corresponding (see Ex. 4) the ranges are in perspective. 

6. Given any two projective ranges of points. Connect any pair of 
corresponding points and take any two points and O on the connect 
ing line. With as a center construct a pencil of lines in perspective 
with the first range, and with as a center construct a pencil of lines 
in perspective with the second range. Prove by use of Ex. 4 that the 
two pencils are in perspective. Hence show how corresponding points 
of two ranges can be found if three pairs of corresponding points are 
known or assumed. 

7. Given two projective pencils of lines. Take the point of inter 
section of two corresponding lines and through it draw any two lines 
o and o . On o construct a range of points in perspective to the first 
pencil of lines and on o construct a range of points in perspective to 
the second pencil of lines. Prove by use of Ex. 5 that the two ranges 
are in perspective. Hence show how corresponding lines of two pro 
jective pencils can be found if three pairs of corresponding lines are 
known or assumed. 

17. Other one-dimensional extents. We have taken as an example 
of a one-dimensional extent of points the range, or pencil, consist 
ing of all the points on a straight line. It is obvious, however, that 
this is not the only example of a one-dimensional extent of points. 



24 ONE-DIMENSIONAL GEOMETRY 

In fact, any curve, whether in the plane or in space, is a one- 
dimensional extent, the coordinate of an element of which may be 
denned in a variety of ways. One of the simplest methods is to 
take the length of the curve measured from a fixed point to a vari 
able point as the coordinate of the latter point, but other methods 
will suggest themselves to the reader familiar with the parametric 
representation of curves. In the case of a circle, for example, we 
may construct a pencil of lines with its vertex on the circle, take 
as the initial line of the coordinate system the tangent line to the 
circle through the vertex of the pencil, and then take as the coordi 
nate of a point on the circle the coordinate of the line of the pencil 
which passes through that point. 

Similarly, the tangent lines to a plane or space curve form an 
example of a one-dimensional extent of lines. Also the tangent 
planes to a cone or a cylinder or the osculating planes to a space 
curve are examples of a one-dimensional extent of planes. These 
extents, both of lines and planes, will be discussed later. 

Moreover, it is not necessary that we confine ourselves to points, 
lines, and planes as elements. We may, for example, take the 
circle in a plane as the element of a plane geometry. In that case 
all the circles which pass through the same two points form a one- 
dimensional extent, a pencil of circles. Another example of a one- 
dimensional extent of circles consists of all circles whose centers lie 
on a fixed curve and whose radii are uniquely determined by the 
positions of their centers. 

In like manner the sphere may be taken as the element of a 
space geometry. All the spheres which intersect in a fixed circle 
form then a one-dimensional extent of spheres, a pencil of spheres, 
and other examples are readily thought of. 

In all these cases, when the coordinate X of the element of the 
extent is fixed, the discussion of the previous sections applies. 

One more remark is important. In all cases we have allowed X 
to take complex values. That is, X is a number of the type 



where i = V 1. The variable X may accordingly be interpreted in 
the usual manner on the complex plane. The significance of the 
linear transformation may then be studied from the standpoint of 



PllOJECTIVITY 25 

the theory of functions of a complex variable. This lies completely 
outside of the range of this book. 

We notice, however, that in interpreting X as the coordinate 
of a point on a straight line we have a one-dimensional extent of 
complex values, while in interpreting it as a complex point on a 
plane we have a two-dimensional extent of real values. That is, 
the dimensions of an extent will depend upon whether it is counted in 
terms of complex quantities or of real quantities. Usually we shall 
in this book count dimensions in terms of quantities each of which 
may take complex values. 

Consider the complex quantity 

\ = \ 4- i\, (1) 

where \ and \ are real, and let 

\ =/!(). *,=/,(0. (2) 

t being a real quantity and the functions real functions. 

Then as t varies, the point X traces out a curve on the complex 
plane which is one-dimensional. If X is interpreted as the coordi 
nate of a point on a straight line, then equations (2) define a one- 
dimensional extent of points on the straight line, which do not of 
course contain all the points of the line. Such a one-dimensional 
extent of points is called a thread of the line. Examples are the 
thread of real points (X 2 = 0), the thread of pure imaginary points 
(Xj =0), the thread of points X x (l -h i) the square of whose 
coordinates is pure imaginary, and others which can be formed 
at pleasure. 

REFERENCES 

Students who wish to read more on the subject of projectivities may consult 
the following short texts : 

LING, WENTWORTH, and SMITH, Elements of Protective Geometry. Ginn and 

Company. 

LEHMER, Synthetic Protective Geometry. Ginn and Company. 
/DOWLING, Projective Geometry. McGraw-Hill Book Company, Inc. 

These books differ from the present one in being synthetic instead of analytic in 
treatment, and they go beyond the content of our Part I in discussing two-dimensional 
extents. In spite of that they may easily be read at this point. If larger treatises 
are needed, consult the references at the end of Part II of this book. 



PART II. TWO-DIMENSIONAL GEOMETRJ 
CHAPTER IV 

POINT AND LINE COORDINATES IN A PLANE 

18. Homogeneous Cartesian point coordinates. Let OX and OY 
be two axes of coordinates, which we take for convenience as rec 
tangular. Then, if P is any point and PM is drawn perpendicular 
to OX, meeting it at M, the distances OM and HP, with the usual 
conventions as to signs, are the well-known Cartesian coordinates 
of P. To make the coordinates homogeneous we place 

OM=-, MP = y - (1) 

* t 

Then to any point P corresponds a definite pair of ratios x\y\t. 
Conversely, to any real pair of ratios x : y : t, in which t is not equal 
to zero, corresponds a real point. In order that a point may cor 
respond to any pair of ratios we need to make the following 
definitions, in harmony with the general conventions of 3 and 4 : 

(1) The ratios 0:0:0 shall not be allowable, for they make both 
OM and MP indeterminate, and the point P cannot be fixed. 

(2) Complex ratios shall be said to represent an imaginary 
point ( 3). 

(3) A set of ratios in which t = shall be said to represent a 
point at infinity ( 4). In fact, it is obvious that as t approaches 
zero, P recedes indefinitely from 0, and conversely. In particular, 
the point 0:1:0 is the point at infinity on the line OY ( 7), the 
point 1:0:0 is the point at infinity on the line OX, and a : b : is 

the point at infinity on the line OM= j-MP. 

19. The straight line. It is a fundamental proposition in analytic 
geometry that any linear equation 

Ax + By + Ct=Q (1) 

represents a straight line. This is partly a theorem and partly a 
definition. It is a theorem as far as it concerns real points whose 

27 



28 TWO-DIMENSIONAL GEOMETRY 

coordinates satisfy an equation of the form (1), in which the coeffi 
cients are all real and A and B are not both zero. For proof of the 
theorem we refer to any textbook on analytic geometry. 

The proposition is a definition as far as it refers to imaginary 
points, to equations with complex coefficients, or to the equation 
t 0. In this sense " straight line " means simply the totality of 
pairs of ratios xiy.t which satisfy equation (1). 

In particular, the equation t = is satisfied by all points at 
infinity. Hence all points at infinity lie on a straight line, called 
the line at infinity. 

If one or more of the coefficients of (1) are complex the straight 
line is said to be imaginary. It is interesting to note that an imag 
inary straight line has one and only one real point. To prove this 
let us place in (1) 

A . = + ia B = + ib C=c + ic. 



Then (1) is satisfied by real values of x, y, and t when and only 
when . 7 A 

a i x + l \y + c f = > 






These equations have one and only one solution for the ratios 
x : y : t, and the theorem is proved. Of course the real point may 
be at infinity. 

Consider now any two straight lines, real or imaginary, with the 
equations 



These equations have the unique solution 

x:y:t=B l C t -B t C l :C 1 A 1 -C t A l :A l B 1 -A 1 B l , 

which represents the common point of the two lines. This point is 
at infinity when A 1 B 2 A B l = 0, in which case, as is shown in any 
textbook on analytic geometry, the lines, if real, are parallel. If 
the lines are imaginary they will be called parallel by definition. 
We may say 

Two straight lines intersect in one and only one point. If the lines 
are parallel, the point of intersection is at infinity. 



POINT AND LINE COORDINATES IN A PLANE 29 

If (X Q , ?/ ) is a fixed point on the line (1), we have 

A*-* ) + (y-y )=0; (2) 

whence u u A 

tj_ *j r 

*-*o~ ~^ 

Whether A and B be real or complex quantities, there exists a 
real or imaginary angle 6 such that 

tan0 = -|. 

Then, from equation (2), 



cos 6 sin 6 

By placing these equal ratios equal to r we have, as another 
method of representing a straight line analytically, the equations 

x = x.+ r cos#, 

x Q N 

y = y Q 4- r sin 6. 

These are the parametric equations of the straight line. In them 
# o , y Q , and 6 are constants and r a variable parameter to each value 
of which corresponds one and only one point on the line, and con 
versely. If the quantities involved are all real, the relation between 
them is easily represented by a figure. In all cases 

%) 2 (4) 



and is defined as the distance between the points (#, y) and (X Q , ?/ o ). 

This work breaks down only when A 2 + B*= 0. In that case 

either A=B=Q, and the line (1) is the line at infinity, or equa 

tion (1) takes the form 

viy + C=Q. (5) 

Here we may still place 

tan 6 = i, 

but sin 6 and cos 6 become infinite and equations (3) are impossible. 
In fact, equation (2) becomes 



This shows that the distance between any two points on the 
imaginary lines (5) must be taken as zero. For that reason they 
are called minimum lines. They play a unique and very important 
part in the geometry of the plane. 



30 TWO-DIMENSIONAL GEOMETRY 

EXERCISES 

1. Prove that through every imaginary point goes one and only one 
real line. 

2. Prove that if a real straight line contains an imaginary point it 
contains also the conjugate imaginary point (that is, the point whose 
coordinates are conjugate imaginary to those of the first point). 

3. Prove that if a real point lies on an imaginary line it lies also on 
the conjugate imaginary line (that is, the line whose coefficients are 
conjugate imaginary to those of the first line). 

4. If the usual formula for the angle between two lines is extended 
to imaginary lines, show that the angle between a minimum line and 
another line is infinite -and that the angle between two minimum lines 
is indeterminate. 

5. Given a pencil of lines with its vertex at the origin. Prove 
that if the pencil is projected on itself by rotating each line through 
a constant angle, the fixed points of the projection are the minimum 
lines. 

6. Show that a parametric form of the equations of a minimum line is 



where t is a parameter, not a length. 

20. The circle points at infinity. The circle is defined analyti 
cally by the equation 

a(x 2 + y 2 ) + 2/rf + 2 gyt + c? = 0, (1) 



the form to which equation (4), 19, reduces when X Q , y^ and r 
are constants and (#, y) are replaced by x:y:t. 

If a 3=- 0, the circle evidently meets the line at infinity in the 
two points 1 : i : and 1 : i : 0, no matter what the values of 
the coefficients in its equation. These two points are called the 
circle points at infinity. If a = in (1), the circle contains the 
entire line at infinity and, in particular, the circle points. Hence 
we may say that all circles pass through the two circle points 
at infinity. 

The circle points 1 : i : are said to be at infinity because they 
satisfy the equation t = 0. Their distance from the center of the 



POINT AND LINE COORDINATES IN A PLANE 31 

circle is not, however, infinite. The distance between two points 
with the nonhomogeneous coordinates (#, y) and (X Q , y o ) is 



which can be written in homogeneous coordinates as 



and this becomes indeterminate when x : y : t is replaced by 1 : i : 0. 

This perhaps makes it easier to understand the statement that 
these points lie on all circles. 

If x Q :y Q : t Q is the center of the circle and r its radius, equation (1) 
can be written (compare equation (2)) 

(zt.-x.ty + W,- y?-r*t\?= 0. 
When r = this equation becomes 

(*.-*. ) + (K-y.o =o, (3) 

the locus of which may be described as a circle with center (# , ?/ o ) 
and radius zero. When the center is a real point the circle (3) 
contains no other real point and is accordingly often called a point 
circle. A point circle, however, contains other imaginary points. 
In fact, equation (3) may be written as 

CO*.- *.0 +* <#.- 8,0] [0*.- *.0 - * (y.- 8,0] = . 

which is equivalent to the two linear equations 



each of which is satisfied by one of the circle points at infinity. 
Hence we have the result that a point circle consists of the two 
imaginary straight lines drawn from the center of the circle to the two 
circle points at infinity. 

The distance from the point (# , # ) to any point on either of 
the two lines just described is zero, by virtue of equation (3). 
There are therefore the minimum lines of 19, as is also directly 
visible from equations (4). It is obvious that through any point 
of the plane go two minimum lines, one to each of the circle points 
at infinity. 



32 TWO-DIMENSIONAL GEOMETRY 

EXERCISES 

1. Show that an imaginary circle may contain either no real point, 
one real point, or two real points. 

2. Consider the pencil of circles composed of all circles through two 
fixed points. Show that the pencil contains two point circles and one 
circle consisting of a straight line and the line at infinity. Show also 
that the point circles have real centers when the fixed points of the 
pencil of circles are conjugate imaginary, and that the point circles 
have imaginary centers when the fixed points are real. 

3. If a pencil of circles consists of circles through a fixed point and 
tangent at that point to a fixed line, where are the point circles and 
the straight line of the pencil ? 

21. The conic. An equation of the second degree, 

aa?+ 2 hxy + % 2 + 2fxt + 2gyt + ct*= 0, (1) 

represents a locus, called a conic, which is intersected by a general 
straight line in two points. For the simultaneous solution of the 
equation (1) and the equation 

Ax+By + Ct=Q (2) 

consists of two sets of ratios except for particular values of A, B, 
and C. 

Let the equation (1) be written in the nonhomogeneous form 
by placing = 1, and let (2) be written in the form ( 19) 

x = x Q +rcos0, y = y Q + r sin 6. (3) 

The values of r which correspond to the points of intersection 
of the straight line (2) with the curve (1) will be found by sub 
stituting in (1) the values of x and y given by (3). There results 
Lr 2 +2Mr + N=Q, (4) 

where M = (ax Q + hy Q +/) cos 6 + ( hx Q -f- ly Q + g) sin 6. 

This will be zero for all values of 6 when X Q and y Q satisfy the 
equations r +Ay +/= 0, hx a + by + ff = 0. (5) 

In this case the point (X Q , y Q ~) will be called the center of the 
curve, since any line through it meets the curve in two points 
equally distant from it and on opposite sides of it. Now equation 
(5) can be satisfied by a point not on the line at infinity when 
and only when h 2 ab3=Q. Hence the conic (1) is a central conic 
when h 2 ab= 0, and is a noncentral conic when h 2 ab = 0. 



POINT AND LINE COOKDINATES IN A PLANE 33 

The conic (1) is cut by the line at infinity t = in two points 
for which the ratio x : y is given by the equation 

ax*+2hxy + by 2 = Q. (6) 

This has equal or unequal roots according as h z ab is equal or 
unequal to zero. Hence a central conic cuts the line at infinity in two 
distinct points; a noncentral conic cuts the line at infinity in two 
coincident points. 

So far the discussion is independent of the nature of the coeffi 
cients of (1). If, however, the coefficients are real the classifica 
tion may be made more closely, as follows : 

(1) h 2 ab<0. The curve cuts the line at infinity in two distinct 

imaginary points. It is an ellipse in the elementary sense, or 
consists of two imaginary straight lines intersecting in a real 
point not at infinity, or is satisfied by no real point. 

(2) h 2 ab > 0. The curve cuts the line at infinity in two distinct real 

points. It is a hyperbola or consists of two real nonparallel lines. 

(3) h 2 ab = 0. The curve cuts the line at infinity in two real coin 

cident points. It is a parabola, or two parallel lines, or two 
coincident lines. In the very special case in which h = a = b = 
it degenerates into the line at infinity, and the straight line 
fx + gy + ct = 0. 

EXERCISES 

1. Show that for a given conic there goes through any point, in 
general, one straight line such that the segment intercepted by the conic 
is bisected by the point. 

2. Show that for a given conic there go through any point, in gen 
eral, two lines which have one intercept with the conic at infinity. 

3. Prove that through the center of a central conic there go two 
straight lines which have both intercepts with the conic at infinity. 
These are the asymptotes. Show that the asymptotes of an ellipse are 
imaginary and those of a hyperbola real, and find their equations. 

4. Show from (3) that if X Q : ?/ : t Q is a point on the conic, the equa 
tion of the tangent line is 

0*o + 7 % + A) x + ( ; ^ + % + 0* ) y + (A + W* + c *o) * = - 

5. Show that the condition that (1) should represent straight lines is 

a h f 

h I g = 0. 



34 



TWO-DIMENSIONAL GEOMETKY 



22. Trilinear point coordinates. Let AB, BC, and CA (Fig. 8) 
be three fixed straight lines of reference forming a triangle and let 
k^ k z , and k 3 be three arbitrarily assumed constants. Let P be any 
point in the plane ABC and let p^ jt? 2 , and p z be the three perpen 
dicular distances from P to the three lines of reference. Algebraic 
signs are to be attached to each of these distances according to 
the side of the line of reference on which P lies, the positive side 
of each line being assumed at 
pleasure. 

The coordinates of P are 
defined as the ratios of three 
quantities # , # , x s such that 



(1:0:0) 



It is evident that if P is given, 
its coordinates are uniquely de 
termined. Conversely, let real 
ratios a 1 : a 2 : a s be assumed for 
x^x^.x^. The ratio xjx^aja^ 

furnishes the condition *-= con- 




FIG. 8 



stant, which is satisfied by any 
point on a unique line through A. Similarly, the ratio x z : x 3 = a 2 : a z 
is satisfied by any point on a unique line through C. If these lines 
intersect, the point of intersection is P, which is thus uniquely 
determined by its coordinates. 

In case these two lines are parallel we may extend our coordi 
nate system by saying that the coordinates a l : a 2 : a s define a point 
of infinity. These are, in fact, the limiting ratios approached by 
x \ : x i : x a as ^ recedes indefinitely from the lines of reference. 

We complete the definition of the coordinates by saying that 
complex coordinates define imaginary points of the plane, and the 
coordinates 0:0:0 are not allowable. 

The coordinates of A are 0:0:1, those of B are 0:1:0, and 
those of C are 1:0:0. The ratios of k^ & 2 , and & 3 are determined 
when the point with the coordinates 1 : 1 : 1 is fixed. This point we 
shall call the unit point, and since the k s are arbitrary it may be 
taken anywhere. Hence the coordinate system is determined by three 
arbitrary lines of reference and an arbitrary unit point. 



POINT AND LINE COORDINATES IN A PLANE 35 




The trilinear coordinates contain the Cartesian coordinates as a 
special limiting case, in which the line BC is the line at infinity. If 
BC recedes indefinitely from A, p % becomes infinite, but the factor 
& 3 can be made to approach zero in such a way that Lim ^ s /? s = l. 
(There is an exception only 
when P is on the line BC and 
remains there as BC becomes 
the line at infinity ; in this 
case &J? 3 = 0.) If in addition 
we place ^=^ = 1, the coor 
dinates x l : # 2 : x s become the 
coordinates x\y\t of 18. 

23. Points on a line. If 

y\ : y* : y* and z \ : z * : z * are two 

fixed points, the coordinates of 
any point on the straight line 
joining them are y^+ \z l : y^ -f 
\2 2 : y^ + \2 3 , and any point with these coordinates lies on that line. 

To prove this let Y and Z (Fig. 9) be the two fixed points and P 

Yp 
any point on the straight line YZ. Place - = m. Then, if jt?J, p v 

P Zi 

and p( are the perpendiculars from I 7 , P, and Z respectively on 
AB, it is evident from similar triangles that 



whence 
Similarly, 



From (1), 22, p. ^i pj=jjp, pJ s-_J, 

where /a, /o ,*and /a" are proportionality factors. By substitution 
we have mp n mp rr mp u 

which is the required form, where X = ~- 



FIG. 9 



= p[+ m p l 

Pl 




36 TWO-DIMENSIONAL GEOMETRY 

The above proof holds for any real point P. Conversely, any 
real value of X determines a real m (the coordinates of Y and Z 
being real) and hence determines a real point of P. For complex 
values of X or for imaginary points Y and Z the statement at the 
beginning of this section is the definition of a straight line. 

It is to be noticed that X is an example of the kind of coordinates 
of the points of a range which was discussed in 8. 

24. The linear equation in point coordinates. A homogeneous 
equation of the first degree, 



represents a straight line, and conversely. 

To prove this theorem it is necessary to show that the linear 
equation is equivalent to the equations of 23. Let us have given 



and let y l : y^ : / 3 and z l : z 2 : z & be two points on the locus of (1). 

Then 



From these three equations we have 

-0. 

Then from the theory of determinants there exist three multi 
pliers \ 1? X 2 , X 3 such that 



whence ^ : a? 2 : ^ 3 = ^ + X^ : y 8 + Xz 2 : y a + X^. 

Conversely, if equations of the form (2) are given ^ve may write 
themas 



POINT AND LINE COORDINATES IN A PLANE 37 
The elimination of p and X then gives 



which is a linear equation in x^ # 2 , and x s . 

Hence equation (1) is equivalent to equation (2), and the 
theorem at the beginning of this section is proved. 

25. Lines of a pencil. If 

= 0, (1) 



are two fixed lines, the equation of any line through their point of 
intersection is 



It is evident that (3) represents a straight line and that the 
coordinates of any point which satisfy (1) and (2) satisfy also (3). 

Furthermore, X is uniquely determined by the coordinates of any 
point not 011 (1) and (2). Hence for all values of X, (3) defines 
the lines of a pencil. 

The parameter X in (3) is of the type of coordinates defined in 
10. To show this let us take Y (y^i y z : y g ), a point on (1), and 



point on (3) and also a point of the range determined by Y and Z. 
By 9, X is the coordinate of a point on the range, and hence, as 
shown in 16, the coordinate of a line of the pencil in the sense 
of 10. 

EXERCISES 

1. Show that the equation of any line through the point A of the 
triangle of reference is x l + A# 2 = 0, and find the coordinates of the 
point in which it intersects any line a^ + a 2 x 2 + a^x & = 0. Distinguish 
between the cases in which g =. and 3 = 0. 

2. Write the equations of two projective pencils of lines with 
the vertices A and B respectively. Find the equation satisfied by 
the coordinates of the points of intersection of corresponding lines. 
Hence verify Ex. 1, 16. 

3. Write the coordinates of the points of two projective ranges on 
AB and AC respectively. Find the equations of the lines connecting 
corresponding points. Hence verify Ex. 2, 16. 



38 TWO-DIMENSIONAL GEOMETRY 

4. Show that homogeneous point coordinates are connected by the 

relation 

p (ak^ + bkjc 2 + ck a x a ) = K, 



where a, b, and c are the lengths of the sides of the triangle of reference 
and K is its area. Hence show that 



bk 2 x 2 + ck a x s = 
is the equation of the straight line at infinity. 

5. Consider the case in which B is at infinity, A and C are right 
angles, and k : = k 2 = k s = 1. Show, for example, that a^ + # 3 is 
the equation of the straight line at infinity and that x 1 + x s -\- Xx 2 = 
is the equation of any straight line parallel to A C. 

26. Line coordinates in a plane. The coefficients a^ 2 , 3 in the 
equation of a straight line are sufficient to fix the line. In fact, 
to any set of ratios a l : 2 : a s corresponds one and only one line, 
and conversely. These ratios may accordingly be taken as coor 
dinates of a straight line, or line coordinates, and a geometry may 
be built up in which the element is the straight line and not 
the point. 

A variable or general set of line coordinates we shall denote by 
u^.u^.u^, and the line with these coordinates is the straight line 
which has the point equation 



This equation may also be considered as the necessary and suffi 
cient condition that the line u^ : u 2 : u 3 and the point x^ix^i x z are 
" united " ; that is, that the point lies on the line and the line 
passes through the point. 

It is obvious that the definition of line coordinates holds for 
Cartesian as well as for trilinear coordinates. With the use of 
trilinear coordinates any straight line may be given the coordinates 
1:1:1. For the substitution 

x( x , x( 

-? pX * = a: pX *=a; 

which amounts to a change in the constants k^ & 2 , k z in (1), 
22, changes the equation a^-f- # 2 ^ 2 + a 3 x s mto the equation 



POINT AND LINE COORDINATES IN A PLANE 39 

27. Pencil of lines and the linear equation in line coordinates. If 

v i : v z : v s anc ^ w i : W 2 : w s are ^ wo nxe d lines, it follows immediately 
from 25 that 

v l 4- \w l : v 2 4- \w z : v s 4- Xw 3 (1) 

represents any line of the pencil determined by the two lines v { 
and w? 

Consider now an equation of the first degree in line coordinates, 

l*l+ a " a +S M 8=- (2) 

It may be readily shown, as in 24, that if v l : v 2 : v 3 and w l : w^ : w s 
are two sets of coordinates satisfying (2), the general values of 
u^.u^: u^ which satisfy (2) are of the form (1). Hence (2) repre 
sents a pencil of lines. 

Or we may argue directly from (1), 26, and say at once that 
any line whose coordinates satisfy (2) is united with the point 
a l : a 2 : 3 and, conversely, that any line united with the point t : & 2 : a 3 
has coordinates which satisfy (2). We have, therefore, the theorem : 

Tlie equation a^ + a z u 2 4- a & u 3 = represents a pencil of lines of 
which the vertex is the point a^ : a 2 : a s . 

Compare the linear equation in point coordinates, 



and the linear equation in line coordinates, 



Equation (3) is satisfied by all points on a range of which the 
base is the line with the line coordinates a l : a 2 : a 3 . It is the point 
equation of that line. 

Equation (4) is satisfied by all lines of a pencil of which the 
vertex is the point with the point coordinates a 1 :a a : 8 . It is 
the line equation of that point. 

EXERCISES 

1. If ABC is the triangle of reference, as in Fig. 8, show that the 
line coordinates of AB are 1:0:0, those of EC are 0:0:1, and those of 
CA are 0:1:0. Show also that the equation of the point A in line 
coordinates is w 8 = 0, that of B is u z = 0, and that of C is u^ = 0. 

2. What does the equation w 1 4-Xw 2 = represent? What line is 
represented by the line coordinates X : 1 : ? 



40 TWO-DIMENSIONAL GEOMETEY 

3. Find in line coordinates the equations of the points of the range 
which lie on the line 1:1:1; also the point coordinates of the same 
range. 

4. Find in point coordinates the equations of the lines of the pencil 
with vertex 1:1:1. Find also the line coordinates of the lines of the 
same pencil. 

5. Show that line coordinates are proportional to the segments cut 
off by the line on the sides of the triangle of reference, each segment 
being multiplied by a constant factor. 

6. Show that line coordinates are proportional to the three perpen 
diculars from the vertices of the triangle of reference to the straight 
line, each perpendicular being multiplied by a constant factor. 

28. Dualistic relations. The geometries of the point and the line 
in a plane are dualistic (2). This arises from the fact that the 
algebraic analysis is the same in the two geometries. The differ 
ence comes in the interpretation of the analysis. In both cases we 
have the two independent ratios of three variables which are used 
homogeneously. In the one case these ratios are interpreted as the 
coordinates of a point; in the other case they are interpreted as 
the coordinates of a line. In both cases we have to consider a 
linear homogeneous equation connecting the variables which is sat 
isfied by a singly infinite set of ratio pairs. In the point geometry 
this equation is satisfied By the singly infinite set of points which 
lie on a straight line. In the line geometry this equation is satis 
fied by the singly infinite set of straight lines which pass through 
a point. 

From the above it appears that any piece of analysis involving 
two independent variables connected by one or more homogeneous 
linear equations has two interpretations which differ in that " line " 
in one is " point " in the other, and vice versa. Hence a geometric 
theorem involving points and lines and their mutual relations may 
be changed into a new theorem by changing point " to " line " and 
" line " to " point." In making this interchange, of course, such 
other changes in phraseology as will preserve the English idiom 
are also necessary. For example, " point on a line " becomes " line 
through a point," and " a line connecting two points " becomes " a 
point of intersection of two lines." 



POINT AND LINE COORDINATES IN A PLANE 41 



We restate some of the results thus far obtained in parallel 
columns so as to show the dualistic relations. 



The ratios x 



x 8 determine 



a point. 

A linear equation aft 4- 
a 8 x 3 = represents all points on 
the line of which the coordinates 
are a l : a >2 : a^. It is the equation of 
the line. 

If y { and z { are fixed points the 
coordinates of any point on the 
line connecting them are ?/ t . -f- Az t . 

If aft + aft + aft = 
and b 1 x l -f- I 2 x 2 + bft = 



The ratios u l : u^ : u s determine 
a straight line. 

A linear equation a^ -f- a 2 u 2 -\- 
a s u 8 = represents all lines through 
the point of which the coordinates 
are a l : a^ : a & . It is the equation of 
the point. 

If v i and w f are fixed lines the 
coordinates of any line through their 
point of intersection are v { + Av 



If 



and 



^ = 
u = 



are the equations of two lines, the are the equations of two points, 
equation of any line through their the equation of any point on the 
point of intersection is line connecting them is 



Three points y iy z t , t { lie on a Three lines v it w i} u { meet in 



straight line when 



a point when 



y\ z 

2/ 2 *, 

2/3 * 



= 0. 



= 0. 



Three straight lines 
meet in a point when 



Three points 
lie on a straight line when 



= 



a * b * C 2 



= 0, 



2 2 

fc. c. 



= 0. 



29. Change of coordinates. We will first establish the relation 
between a set of Cartesian coordinates and a set of triliiiear coor- 
^dinates. Let AB, EC, and CA be the lines of reference of the 



42 TWO-DIMENSIONAL GEOMETBY 

trilinear coordinates and let their equations referred to any set of 
Cartesian coordinates be respectively 



(1) 



Then by a familiar theorem in analytic geometry, 
= q^ 4 6 a y + g^ 



= 



= 



We may take without loss of generality 



since each of the equations (1) may be multiplied by a factor 
without changing the lines represented. 

Therefore we have 

px^a^ + b^ + cj, 

px^a^x + tyi-cf, (2) 



where /> is a proportionality factor. 

Since the lines ^4.#, ^(7, and (7^4 form a triangle, the determinant 
I a J ) 2 C s ^ oes no ^ van i s ^ an d equations (2) may be solved for x, y, 
and t. 

Suppose now another triangle A f B C f be taken, the equations of 
its sides being 

4*+ty+4*A 

+ 4t=Q, (3) 



and let x[: x 2 : x s be trilinear coordinates referred to the triangle 
A B C . Then, as before, 

p x[ = a[x + % + ^I. 

/>X = 4* + &;y + * (4) 

pX = 8 2: + ijy 4- cj. 



POINT AND LINE COORDINATES IN A PLANE 43 

Equations (2) may be solved for x, y, and t and the results 
substituted in (4). There result relations of the form 



fa* 



which are the equations of transformation of coordinates from 
x l : x 2 : x s to 4 : x 2 : x(. 

In (5) the right-hand members equated to zero give the equa 
tions in trilinear coordinates of the sides of the triangle of reference 
A B C . Since these do not meet in a point the coefficients are sub 
ject to the condition that their determinant does not vanish, and 
this is the only condition imposed upon them. 

By the transformation (5) the equation of the straight line 

?Vi+ V 2 +V 3 =0 
becomes u(x{ + u 2 x 2 + u ^x( = 0, 

where = X + + 



These are the formulas for the change of line coordinates. 

In connection with the change of coordinates three theorems are 
of importance. 

/. The degree of an equation in point or line coordinates is unaltered 
by a change from one set of trilinear coordinates to another. 

II. If the coordinates y i and z { are transformed into the coordinates 
y( and 2 t , the coordinates y { + \z f are transformed into the coordinates 
y[ + X fcJ, where \ f = c\, c being a constant. 

III. The cross ratio of four points or four lines is independent of 
the coordinate system. 

Theorem I follows immediately from the fact that equations (5) 
and (6) are linear. 

To prove theorem II note that from (5), if the coordinates 
\z t . are transformed into x { , then 



44 



TWO-DIMENSIONAL GEOMETRY 




where & i and <r 2 are used, since in transforming y { and z. by (5) the 
proportionality factors may differ. 

Similar expressions may be found for x 2 and x f 3 . Hence we have 

x l : x r 2 : x( = y( + \z{ : y 2 -\- \z! 2 : y( + \z(, which proves the 

"i "i /i 

theorem. The same proof holds for line coordinates using equa 
tions (6). 

Theorem III follows at once from II. 

30. Certain straight-line configurations. A complete n-line is 
denned as the figure formed by n straight lines, no three of which 
pass through the same point, together 
with the \ n (n 1) points of inter 
section of these lines. A complete 
three-line is therefore a triangle con 
sisting of three sides and three vertices. 
A complete four-line is called a com 
plete quadrilateral and consists of four 
sides and six vertices. Thus in Fig. 10 
the four sides are a, 5, <?, d and the six 
vertices are K, L, M, .2V, P, Q. Two 

vertices not on the same side are called opposite, as K and M, L 
and JV, P and Q. A straight line joining two opposite vertices is a 
diagonal line. The complete quadrilateral has three diagonal lines. 

A complete n-point is de 
fined as the figure formed by 
n points, no three of which lie 
on a straight line, together 
with the ^n(n 1) straight 
lines joining these points. A 
complete three-point is there 
fore a triangle consisting of 
three vertices and three sides. 
A complete four-point is called 

a complete quadrangle and 

. / , , FIG. 11 

consists of four vertices and 

six sides. Thus in Fig. 11 the four vertices are A, B, C, D and 
the six sides are &, ?, w, w, p, q. Two sides not passing through the 
same vertex are called opposite, as k and m, I and rc, and p and q. 



FIG. 10 




POINT AND LINE COORDINATES IN A PLANE 45 



The point of intersection of two opposite sides is a diagonal 
point. The complete quadrangle has three diagonal points. 

It is obvious that a complete %-point and a complete n-line are 
dualistic. A triangle is dualistic to a triangle, and a complete 
quadrangle to a complete quadrilateral. The diagonal lines of 
a complete quadrilateral are dualistic to the diagonal points of a 
complete quadrangle. 

For the complete triangle we shall prove the following dualistic 
theorems : 

L The theorem of Desargues. If two triangles are so placed that the 
straight lines connecting homologous vertices meet in a point, then the 
points of intersection of homologous sides lie on a straight line. 

II. If two triangles are so placed that the points of intersection 
of homologous sides lie on a straight line, then the lines connecting 
homologous vertices meet in a point. 

Let there be given two triangles with the vertices A, B, C and 
A 1 , B , C respectively (Fig. 12) and with the sides a, b, c and 
a , b , c respectively, the 
side a lying opposite the 
vertex A etc. 

We shall denote by 
AA the straight line 
connecting A and A , 
and by aa the point 
of intersection of a and 
a . Then the two the 
orems stated above are 
respectively : 



(MO 




If the straight lines 9 

"Frr 19 

AA ,BB , and CC meet 

in a point 0, the points aa , bb r , and cc f lie on a straight line o. 
If the points aa 1 , bb f , and cc f lie on a straight line o, the straight 
lines AA , BB , and CC meet in a point 0. 

The proofs of these theorems may be given together, the upper 
line of the following sentences being read for theorem I and the 
lower line for theorem II. 



46 TWO-DIMENSIONAL GEOMETRY 

Take K _ > as triangle of reference and ! \ as the unit 
. I abc J UJ f 

<Y f Then the coordinates of ! | are 0:0:1, those of *! i f 
are 0:1:0, those of f**\ are 1:0:0, and those of f \ are 1:1:1. 

UJ r^n UJ r/n 

By 28 the coordinates of -j , > are 1 : 1 : 1 + \, those of 4 7) V 

are 1 : 1 + /i : 1, and those of j , r are 1 + i/ : 1 : 1. 

The coordinates of any / P oint on A B \ are therefore 

lline through a b i 

-i , i i /> i v i i <* j i.1.- f point lies also on AB ~\ 

l + /3:l + /3(l+/A):l-hX-f /o, andif this \ b , ., u xT 

^passes also through abj 

we must have /} = !. Hence the coordinates of -! f \- are 

{f)f) f 1 ^CC J 

* are z^ : : \ and 
BB J 

the coordinates of "I jj/j are v: JJL: 0. Since 



-/* 
v 

V U 



= 0, 



i.v j.1. f points aa f , W, cc f \ , f line o ") ,~, 

the three { ,F ., __. _. r have a common -j . , -* : >. The 

Limes AA t BB\ CC J Lp omt J 

two theorems are therefore proved. 

rp, f pointl . . , f line o "} . 

The < r v J- equation of the -J . , > is 
L line J \pomt Oj 

C \IJLX -f~ ^A,# -f- ffa? = "1 

J I 2 3 I . 



For the complete quadrilateral we shall prove the following 
theorem : 

///. Any two diagonals of a complete quadrilateral intersect the 
third diagonal in two points which are harmonic conjugates to the two 
vertices which lie on that diagonal. 

In Fig. 13 let the two diagonals LN and MK intersect the third 
diagonal PQ. in the points R and S respectively. We are to prove 
that R and S are harmonic conjugates to P and Q. 

Since by III, 29, the cross ratio is independent of the coordi 
nate system, we shall take the triangle LPQ as the triangle of 




POINT AND LINE COORDINATES IN A PLANE 47 

reference and the point N as the unit point, so that the coordi 
nates of P are 0:0:1, those of Q are 1:0:0, those of L are 
0:1:0, and those of N are 1:1:1. Then by 23 it is easy 
to see that the coordinates of 
R are 1:0:1, those of M are 
0:1:1, those of Tfare 1 : 1 : 0, 
and finally that those of S 
are -1:0:1. By 14 the 
theorem follows. 

The dualistic theorem to III 
is as follows: 

IV. If any two diagonal points p^ "R g V 
of a complete quadrangle are p IG ^3 f\ j> &\ 
joined by straight lines to the 

third diagonal point, the two joining lines are harmonic conjugates 
to the two sides of the quadrangle which pass through that third 
diagonal point. 

The proof is left to the reader. 

Since the cross ratio of any four lines of a pencil is equal to 
the cross ratio of the four points in which the four lines cut any 
transversal ( 16), theorem IV leads at once to the following: 

V. The straight line connecting any two diagonal points of a com 
plete quadrangle meets the sides of the quadrangle which do not pass 
through the two diagonal points, in two points which are harmonic 
conjugates to the two diagonal points. 

Similarly, theorem III may be replaced by the theorem, dualistic 
to V, as follows: 

VI. If the intersection of any two diagonal lines of a complete 
quadrilateral is connected with the two vertices of the quadrilateral 
which do not lie on the two diagonals, the two connecting lines are 
harmonic conjugates to the two diagonals. 

Theorem III gives a method of finding the fourth point in a 
harmonic set when three points are known. In Fig. 13 let us 
suppose P, Q, and R given, and let it be required to find S. The 
point L may be taken at pleasure and the lines LP, LR, and LQ 
drawn. Then the point N may be taken at pleasure on LR and 



48 



TWO-DIMENSIONAL GEOMETRY 



the points M and K determined by drawing QN and PN. The 
line MK can then be drawn, determining S. 

We will now prove the following theorem : 

VII. Theorem of Pappus. If P^ P%, P b are three points on a 
straight line and P 2 , P, P & are three points on another straight line, 
the three points of intersection of the three pairs of lines PP 2 and 
PtP b , JJJJ and P b P^ P Z P and P^ lie on a straight line. 

We may so choose the coordinate system that the line contain 
ing JJ, P 8 , Pr, (Fig. 14) shall be x 1 = and the line /containing 
JJ, 7^, 7J shall be # 2 = 0. We may then take the line P 1 P 2 as the 
line X B = 0, so that the coordi 
nates of P 1 are (0:1: 0) and 
those of P z are (1:0: 0), and 
may so take the unit point 
that the coordinates of P & are 
(0:1:1) and those of JJ are 
(1:0: 1). Call the coordinates 
of P b (0 : 1 : X) and those of 
P 6 (1 : : fj,). Then the equa 
tion of J^P 2 is x 3 = and that 
of PiP b is 2^+ \x 2 - x s = 0. These 
lines intersect in the point 
K (X : -1 : 0). The equation 










is x z x s =Q and that 



FIG. 14 



= 0, 



of 

of P^PQ is At^+ Xz 2 x 3 = 0. These lines intersect in the point 
L (1 X : IJL : IJL*). The equation of P S P 4: is x^+ x^ x 3 = and that of 
is /Jt X^ x 3 = 0. These lines intersect in M (1 : p 1 : JJL). Since 

X -10 

1 - X fj, /* 

1 fJL-l 

the three points L, K, M lie in a straight line, as was to be proved. 

Dualistic to this theorem is the following : 

VIIl. If p^ jt? 3 , p 5 are three straight lines through a point and 
p 2 , p^, p 6 are three straight lines through another point, the three lines 
connecting the three pairs of points jt?j? 2 and 

PsPt an d P&P\ mee t ^ n a point- 
The proof is left to the reader. 



and 



POINT AND LINE COORDINATES IN A PLANE 49 

EXERCISES 

1. Prove theorem IV. 

2. Prove theorem VIII. 

3. A triangle is so placed that its vertices P, Q, R are on the sides 
AB, AC, and BC, respectively, of a fixed triangle and its sides PR and 
RQ pass through two fixed points in a straight line with A. Prove that 
the side PQ passes through a fixed point. 

4. A triangle is so placed that its sides QR, PR, PQ pass through 
the vertices C, B, A, respectively, of a fixed triangle and its vertices Q 
and P lie on two fixed lines which intersect on BC. Prove that the 
vertex R lies on a straight line. 

5. Given a straight line^? and two fixed points A and.S. Take any 
two points on p and connect each of them with A and B. These lines 
determine two new points C and D by their intersections. Prove that 
the line CD passes through a fixed point on AB. 

6. Given a point P and two fixed lines a and b. Draw any two lines 
through P and connect their points of intersection with a and b. This 
determines two new lines c and d. Prove that the point of intersection 
of c and d lies on a fixed straight line through ab. 

7. Three lines/, g, h are drawn through the vertex A of the triangle 
ABC. On g any point is taken and the lines I and m are drawn to C 
and B respectively. The line I intersects f in D and the line m inter 
sects h in E. Prove that DE passes through a fixed point on BC. 

8. Three points F, G, H are taken on the side BC of. the triangle 
ABC. Through G any line is drawn cutting AB and AC in L and M 
respectively. The lines FL and HM intersect in K. Prove that the 
locus of K is a straight line through A. 

9. Show that if a, a and b, b are any two pairs of corresponding 
lines of two projective pencils not in perspective, the line connecting 
the points ab and a b passes through a fixed point. This is called the 
center of homology of the two pencils. Show that it is the intersection 
of the two lines which correspond to the line connecting the vertices of 
the pencils, considered as belonging first to one pencil and then to the 
other. 

10. Show that if A, A and B, B are any two points of two projec 
tive ranges which are not in perspective, the point of intersection of 
the lines AB and A B lies on a fixed straight line. This is called the 
axis of homology of the two ranges. Show that it intersects the base of 
each range in the point which corresponds to the point of intersection 
of the two bases, considered as belonging to the other range. 



50 TWO-DIMENSIONAL GEOMETRY 

31. Curves in point coordinates. The equations 



where t is an independent variable and the ratios of the functions 
t () are not constant or indeterminate, define a one-dimensional 
extent of points called a curve. It is not necessary that any point 
of the curve should be real. We shall limit ourselves to those 
curves for which the functions $ t -(T) are continuous and have 
derivatives of at least the first order. 

If 3 (T) is identically zero the curve is the straight line x z = 0. 
Otherwise we may write equations (1) in the form 

5 = &( = * (0 , s = MQ = j. (0 . (2) 

x, <f>,(0 x <j> CO 

3 *3Vx 3 f-9\.S 

It is then possible to eliminate t between the equations (2) with 
the result, 

(3) 



Conversely, let there be given an equation 

/Ov*v*,)=0, (4) 

where /is a homogeneous function in x^ x^ x s . By a homogeneous 
function we mean one which satisfies the condition 



where \ is any multiplier, not zero or infinity. In particular, if we 
place \= WQ have 



for all points for which x 3 is not zero. Equation (4) may then be 
written /(, U)=0, (5) 



g, 





where 



We shall limit ourselves to functions /which are continuous and 
have partial derivatives of at least the first order. 

We shall also assume that (4) is satisfied by at least one point 



: Ml} 



one ^ ^ ne P ar tial derivatives / say ~ \ 



which is valid in the vicinity of t=, s= 

y, 

This last equation may be written 



POINT AND LINE COORDINATES IN A PLANE 51 

does not vanish. Then similar conditions hold for (5), and by the 
theory of implicit functions * we have, from (5), 





y, 



which is of the type of equations (1). Hence, under our hypotheses 
equation (4) represents a curve. 

The above discussion leaves unconsidered the points for which 
x 3 = 0. These may be found by direct-substitution in (4) or we may 
repeat the discussion, dividing by some other coordinate, perhaps x^ 

Let P (j/j : y 2 : y^) be a point of (1) corresponding to the value 
t = o , and let Q (y l + ky l : y^ + ky^ : y z + A?/ 3 ) be a point correspond 
ing to t Q + A. These two points fix a straight line with the equation 



the coefficients of which are determined by the two equations 



i) + 2 G/ 2 + A # 2 ) + , G/3 

From these it follows that 



It is to be noticed that these involve the ratios of the in 
crements A^, A?/ 2 , Ay 3 . If now A approaches zero, the point 
Q approaches P, the ratios &y l : A?/ 2 : Ay 3 approach the ratios 
dy l : dy 2 : dy^ and the ratios a l : a 2 : a s approach the limiting ratios 



The straight line (6) with the coefficients defined by (8) is 
the limit of the secant PQ and is called the tangent to the curve. 

If the equation of the curve is in the form (4), the equation of 
the tangent may be modified as follows : 

Since /(y^y^ -y^) is a homogeneous function we have, by 
Euler s theorem, 



See WilymLs " Advanced Calculus. " p. 117. 



52 TWO-DIMENSIONAL GEOMETRY 

On the other hand, dy^ dy^ dy & satisfy the condition 



Equations (9) and (10) give 

P / p/> Qf 

yM- y**y* : vAt*- y^y* : yy*- y^y^ ^ : ^ : ^ 

Hence the equation of the tangent line is, from (8) and (10), 



The equation (11) is fufly and uniquely determined for any 
point on the curve except for a point y l : y^ : y^ at which 

f-=0, = 0, f = 0. (12) 

fy ty t ty t 

Points for which the conditions (12) hold are called singular points. 
We may sum up as follows : At every nonsingular point (y : y^ : y^) 



there is a definite tangent line given by the equation 



Consider now any straight line determined by two fixed points 
y. and z- so that y { + \z { is any point of the line. The point y { + \z ; 
lies on the curve (1) when X has a value satisfying the equation 

/(y,+ H- y,+ H .+ H) = . C 13 ) 

which expands by Taylor s theorem into 

2 \ 2 +... = 1 (14) 



where A= and A=++- 



If y i is on the curve (4), A Q = and one root of (14) is zero. If, 
in addition, A^= and y i is not a singular point, z i lies on the tan 
gent line to (4) and two roots of (14) are zero. If y i is a singular 
point of the curve, A Q = and A^= for all values of 2; that is, 
any line through a singular point of a curve intersects the curve in at 
least two coincident points. 



POINT AND LINE COOKDINATES IN A PLANE 53 



, # 2 , # 8 ) is a homogeneous polynomial of the nth degree, 
the locus of points satisfying (4) is denned as a curve of the nth 
order. Equation (14) is then an algebraic equation of the nth 
degree unless its left-hand member vanishes identically for all 
values of X. Hence any curve of the nth order is cut by any straight 
line in n points unless the straight line lies entirely on the curve. 
32. Curves in line coordinates. The equations 

f ^VV^iCO ^CO ^CO. C 1 ) 

where t is an independent variable and the ratios of the functions 
<f>i(t) are not constant or indeterminate, define a one-dimensional 
extent of straight lines. We shall see that these lines determine 
a curve in the sense of 31. Equations (1) are called the line 
equations of that curve. 

Proceeding as in 31 with the same hypotheses as to the nature 
of the functions </>;(), we may show that equations (1) are 
equivalent to the equation 



Conversely, let there be given an equation 

/(!,,, .)=<> (2) 

where/ is a homogeneous function in u^ w g , u 3 ; we may show, as 
in 31, that equation (2) defines a one-dimensional extent of lines 
of the type (1). 

The discussion now proceeds dualistically to that in 31. 

Let p(y^\ v^i v 8 ) and q(v^ + Ai^: v z + Av 2 : v^ + Ai> 3 ) be two straight 
lines determined by placing t = t Q and t = t Q +At in (1). These 
two lines determine a point K the coordinates of which satisfy the 
two equations ^ + ^ + ^ = , 

( Vl + A Vl X + O 2 + Av 8 X+(i; 8 + &v 3 )x s = 0, 
the solution of which is 



Now let A approach zero. The line q approaches the line p, the 
ratios A^: A^ 2 : A^ 3 approach the ratios dv^i dv^: dv^ and the point 
K approaches the point Z, of which the coordinates are 



54 TWO-DIMENSIONAL GEOMETEY 

By virtue of (3) and (1) the points L form in general a curve. 
An exception would occur when the right-hand ratios of (3) are 
independent of t. In that case the points L for all lines of (1) 
coincide. 

If the extent of lines is defined by a single equation (2) the 
coordinates of L may be put in another form, as follows : Since / 
is a homogeneous function we have, by Euler s theorem, 



But 

whence 



The coordinates of L are therefore 



These equations determine a unique point on any line p unless 
p is such a line that 

J..o, l-o, 1=0, 

PVj <7^ 2 d7V g 

in which case p is called a singular line. 

Equations (4) also show that the points L form a curve unless 

ftf 
the ratios of the partial derivatives are constant in the neigh- 

dv { 

borhood of v t . This would happen, for example, if 
/= (^^4- a w s + a.Mj)^^, u ^ WB ) 

and v t . is any point which makes the first factor vanish. The points 
L on all lines in the neighborhood of v i are then all a^i 2 : a g . 
Leaving the exceptional case aside we have the theorem: 

On any nonsingular line of a one-dimensional extent of lines there 
lies a unique point, called a limit point, the locus of which is in general 
a curve. This curve is said to be defined in line coordinates by the 
equation of the line extent. In special cases the curve may reduce to 
a point or contain a number of points as parts of the curve. 



POINT AND LINE COORDINATES IN A PLANE 55 

In case we have a true curve of limit points it will be possible 
to solve equations (4) for v l : v 2 : v s and substitute in (2). This 

gives /ov <v .) = * <? f *,. *.) = ( 5 ) 

which is the equation in point coordinates of the locus of L. 



From (5), rr = ^^ + jv_p + _v_pj 

di , ^ 2 , 



*^9 o Q P 

^, a.r. a^.y 

where p is a proportionality factor and the last reduction is made 
by means of (4). But since v^+ v z x z + v s x a we n ave 



Therefore -^- = m t .. 

a^ 

This shows that the tangent line to the curve (5) at the point 
L is the line j?. Hence we have the theorem: 

Each line of a one-dimensional extent of lines is tangent at its 
limit point to the curve which is the locus of the limit points. The 
lines therefore envelop the curve. 

Let us suppose now that in equation (2) / is an algebraic poly 
nomial of the wth degree. Then the locus of the limit points L is 
called a curve of the nth class. We shall prove that through any 
point of the plane go n lines tangent to a curve of the nth class. 

To do this we have to show that n lines satisfying equation 
(2) go through any point of the plane. Now any point is fixed 
by two lines v i and w i9 and any line through that point has the 
coordinates t^.+ X^.. This line satisfies (2) when X satisfies the 
equation /( ^ + ^ ^ + ^ ^ + ^ _ Q _ 

This is an equation of the wth degree, and the theorem is proved. 

We have shown in this section that a one-dimensional extent 
of lines are in general the tangent lines to a curve. Conversely, 
the tangent lines to any curve are easily shown to be a one- 
dimensional extent of lines. An exception occurs only when the 
curve consists of a number of straight lines. 



56 



TWO-DIMENSIONAL GEOMETRY 



The dualistic relation between point and line coordinates is 
exhibited in the following restatement, in parallel columns, of the 
results of 31 and 32: 



An equation f(x v x 2 , # 8 ) = is 
satisfied by a one-dimensional ex 
tent of points which lie on a curve. 
A line joining two consecutive 
points of the curve is tangent 
to the curve. Its line coordinates 

elimination of x 1 : x 2 : # 3 between 
these equations and that of the 
curve gives the line equation of 
the curve. 

The equation of the tangent 
line to the curve defined by the 
point extent is 

/. , g /~ , /_ ,, 



An equation f(u lt u 2 , u 8 ) = is 
satisfied by a one-dimensional ex 
tent of lines which are tangent to 
a curve. A point of intersection 
of two consecutive lines is a point 
on the curve. Its point coordinates 



df df 
are x. : x, : x a = -f- : -f- : 



The 



U Z :U B between 



elimination of 
these equations and that of the line 
extent gives the point equation of 
the curve. 

The equation of a point on the 
curve enveloped by the line ex 
tent is 



If / is of the nth degree the 
curve is of the nth order. 

On any line lie n points of the 
curve. 

The curve of the first order is 
a straight line, the base of a pencil 
of points. It is of zero class and 
has no line equation. 



If f is of the nth degree the 
curve is of the nth class. 

Through any point go n lines 
which are tangent to the curve. 

The curve of the first class is 
a point, the vertex of a pencil of 
lines. It is of zero order and has 
no point equation. 



EXERCISES 

1 . Find the singular point of x? + x i x s x x s 0. Show that 
through the singular point go two real lines which meet the curve in 
three coincident points. Sketch the curve with special reference to its 
relation with the triangle of reference. Also sketch the curve interpret 
ing the coordinates as Cartesian coordinates and taking x s = 0, x 2 = 0, 
x l = successively as the line at infinity. 

2. Find the singular point of xf xx s = 0. Show that through it 
go two coincident lines which meet the curve in three coincident points. 
Sketch the curve as in Ex. 1. 



POINT AND LINE COORDINATES IN A PLANE 57 

3. Find the singular point of the curve x? + x?x s + x%x 8 = 0. Show 
that through it go two imaginary lines which meet the curve in three 
coincident points. Sketch the curve as in Ex. 1. 

4. Find the line equation of each of the curves in Exs 1-3. 

5. Show that any point whose coordinates satisfy the three equations 

f = 0, ~ = 0, -^ = lies on the curve f= and is therefore a 
dx^ dx 2 dx 9 

singular point. 

6. Show that the singular points of a curve in nonhomogeneous 

<~\ /> o /, 

Cartesian coordinates are given by ~ = 0, -^- = 0, provided the solu 

tions of these equations also satisfy f(x, y) = 0. (Compare Ex. 5.) 
Apply to find the singular points of x* 4- 2/ 2 = & 2 and x 2 if = 0. 

7. Show that through any point on a singular line of a line extent 
go at least two coincident lines of the extent. Hence show that if the 
extent envelops a curve of the ntli class, the singular lines are the 
locus of a point such that at least two of the n tangents to the curve 
from that point are coincident. Illustrate by considering the line extent 
u} + ujij = 0. 

8. If / (x v o? 2 , x s ) = is the equation of a curve and y^ : ?/ 2 : ?/ 3 is a 
fixed point, show that the equation 



represents a curve which passes through all the singular points of 
/= and through all the points of tangency from y. to /= 0, but 
intersects f = in no other points. 

9. Prove that a curve of the third order can have at most one singu 
lar point unless it consists of a straight line and a curve of second 
order, or entirely of straight lines. 



CHAPTER V 

CURVES OF SECOND ORDER AND SECOND CLASS 

33. Singular points of a curve of second order. By 31 a curve 
of second order is denned by the equation 



which can be more compactly written in the form 

2)0*%==0- (=) 

By the last theorem of 31 any straight line cuts a curve of 
second order in two points or lies entirely on the curve. 

It follows immediately that if the curve has singular points it 
must consist of straight lines. For any line through a singular 
point meets the curve in two points coincident with the singular 
point, and if it passes through a third point of the curve it must 
lie entirely on the curve. 

We proceed to examine the singular points more closely, as 
they are important in determining the nature of the curve. 

By (12), 31, the singular points are the solutions of the 
equations = 



Let Z>, called the discriminant of equation (1), be defined by 

*ii a i 2 a 



There are then three cases in the discussion of equations (2). 

CASE I. Z> = 0. Equations (2) have no solution, and the curve 
has no singular point. This is the general case. 

CASE II. D 0, but not all the first minors of D are zero. 
Equations (2) have one solution, and the curve has one singular 
point. Let that point be taken by a change of coordinates as the 

58 



CURVES OF SECOND ORDER AND SECOND CLASS 59 

point 0:0:1. The degree of the equation will not be changed 
(29), but in the new equation we shall have 13 =0, # 28 =0, 
# 38 =0. The equation therefore becomes 



a u x?+ 2 a^x^ + V 2 2 = 0, 

which can be factored into two linear factors. These factors can 
not be equal, for if they were we should have u : & 12 = 12 : a^, and 
equations (2), written for the new coordinates and new equation, 
would have more than one solution. Hence the locus of (1) con 
sists of two intersecting straight lines. 

CASE III. D= 0, and all its first minors are zero. Any solution of 
one of the equations (2) is a solution of the others, and the curve 
has a line of singular points. If by a change of coordinates that 
line is taken as the line x^= 0, we shall have in the new equation 
a \z~ a is = a zz ~ a = a 38~ an d the equation becomes xf = 0. Hence 
in this case the curve consists of two coincident straight lines. 

Summing up, we have the following theorem : 

A curve of the second order has in general no singular point. If it 
has one singular point it consists of two straight lines intersecting in 
that point. If it has a line of singular points it consists of that line 
doubly reckoned. 

The curves of second order in homogeneous coordinates are the 
same as the conies in Cartesian coordinates, for, as shown in 29, 
the degree of an equation is not altered by a change of coordinates. 
We may on occasion distinguish between the conies without singu 
lar points and those which consist of two straight lines by calling 
the latter degenerate cases of the conic. 

34. Poles and polars with respect to a curve of second order. 
By (11), 31, if y i is a point on the conic (1), 33, the line 
coordinates of the tangent at y i are 



CO 

P U 3= ^1+ a ^+ V* 

Let us now drop the condition that y { is on the curve and consider 
y, as any point of the plane, whether on the curve or not. 



60 TWO-DIMENSIONAL GEOMETRY 

Equations (1) then associate to any point y { a definite line u { . 
This line is called the polar of the point, and the point is called 
the pole of the line. The equation of the polar is 



or, more compactly, 

2} a *Vi x k = 0. (% = %) (2) 

If y. is given, ^ is uniquely determined by (1); but if u t is given, 
y i is determined only when equations (1) can be solved, that is, 
when the discriminant Z>, 33, does not vanish. Hence, 

/. To any point of the plane corresponds always a unique polar; 
but to any line of the plane corresponds a unique pole when and only 
when the curve has no singular point. 

The following theorems are now easily proved : 

II. The polar of a point on the curve is the tangent line at that 
point and, conversely, the pole of any tangent to the curve is the point 
of contact of the tangent. 

It is obvious that equation (2) reduces to the equation of 
the tangent when the point y i is on the curve. Conversely, if 
equation (2) is that of a tangent to the curve, the solution 
of equations (1) will give the point of contact. 

///. The polar of a point passes through the point when and only 
when the point is on the curve. 

This follows from the fact that the substitution nr. = y i reduces 
equation (2) to the equation of the curve. 

IV. The polar of any point passes through the singular points of 
the curve if such exist. 

This follows from the fact that equation (2) can be written 

^(n*i+i^+^ 

V. If a point P lies on the polar of a point Q, then Q lies on the 
polar of P. 



CURVES OF SECOND ORDER AND SECOND CLASS 61 
If P is the point y i and Q is the point ,., the polar of P is 

and that of Q is 

The condition that P should lie on the polar of Q is 



which is just the condition that Q should lie on the polar of P. 

VI. If a curve of second order has no singular point, two tangents 
may be drawn to the curve from any point not on it, and the chord con 
necting the points of contact of these tangents is the polar of the point 
of intersection of the tangents. 

Let P (Fig. 15) be a point not on the curve. The polar of P, 
being a straight line, cuts the curve in two points T and S. These 
two points are distinct because by theorem II the polar is not 
tangent, since P, by hypothesis, is not 
on the curve. 

Since by hypothesis the curve has 
no singular point, it has a unique 
tangent line at each of the points T 
and S. These tangents are the polars 
of their points of contact and hence by 
theorem V pass through P. The polar 
of P therefore passes through T and S 
(theorem V). 

There can be no more tangents 
from P to the curve, for if there were, 

the point of tangency would lie on TS by theorem V, and hence 
TS would intersect the curve in more than two points, which is 
impossible. The possibility that TS should lie entirely on the curve 
is ruled out by the fact that in that case the curve would consist 
of two straight lines and would have a singular point, which is 
contrary to hypothesis. 

This theorem as proved takes no account of the reality of the 
lines and points concerned. In the case in which it is possible to 
draw real tangents from P, however, the theorem furnishes an easy 
method of sketching the polar of P. 




FIG. 15 



62 



TWO-DIMENSIONAL GEOMETRY 



K 



When real tangents cannot be drawn from P, as in Fig. 16, the 
polar of P may be constructed as follows : 

Through P draw two chords, one intersecting the curve in the 
points R and S and the other intersecting the curve in the points 
T and V. Draw the tangents 
at the points R, S, T, and V, 
and let the tangents at R and S 
intersect at L and let the tan 
gents at T and V intersect at K. 
Then, by theorem VI, L is the 
pole of RS, and K is the pole 
of TV. Consequently the polar of 
P passes through L and K and 
is the line LK. 




TIG. 10 



VII. For a curve of second order 
without singular points it is possible 

in an infinite number of ways to construct triangles in which each side 
is the polar of the opposite vertex. These are called self-polar triangles. 

We may take A (Fig. 17), any point not on the curve, and 
construct its polar, which will not pass through A (theorem III) 
and cannot lie entirely on the curve, 
since the curve has no singular point. 
We may then take Z?, any point on 
the polar of A but not on the curve, 
and construct its polar. This polar 
will pass through A (theorem V) but 
not through B (theorem III). The 
two polars now found are distinct 
lines (theorem I) and will intersect 
in a point C. Draw AB. Then AB is 
the polar of C by theorem V. The 
triangle ABC is a self -polar triangle. 

VIII. If any straight line m is passed through a point P, and 
R and S are the points of intersection of m with a curve of the 
second order, and Q is the point of intersection of m with the polar 
of P, then P and Q are harmonic conjugates with respect to R 
and S. 




FIG. 1 



CURVES OF SECOND ORDER AND SECOND CLASS 63 

Let P (Fig. 18) be any point with coordinates y^ let p be the polar 
of P, and let m be any line through P cutting p in Q and the curve 
in R and S. Then, if z. are the coordinates of Q 9 the coordinates of 
R and S are y^X^ and y i +\z i1 where \ and X 2 are the roots of 
the equation 

2, a i 



obtained by substituting x i =y i +\z i in the equation of the curve. 




FIG. 18 



But since Q is on the polar of P, we have 2}%^= ^ 
therefore X t = X a . By 14 the theorem is proved. 

This theorem gives a method of finding the polar of P when 
the curve of second order consists of two straight lines intersecting 
in a point (Fig. 19). Draw through P any straight line m inter 
secting the curve in the points R and S, distinct from 0, and find 
the point Q, the harmonic conjugate of P with respect to fi and S. 
By theorem VIII, Q is on the polar of P, and by theorem IV the 
polar of P passes through 0. Hence Q and determine the re 
quired polar p. 

EXERCISES 

1. Prove that if a conic passes through the vertices of the triangle 
of reference its equation is c^x^ c^a^-f- cjr.jr^= 0. Classify the conic 
according to the nature of the coefficients c,-. 

2. Prove that if the triangle of reference is composed of two tan 
gents to a conic and the chord of contact, the equation of the conic is 
c i x i x s + C 2 X % = Classify the conic according to the nature of the 
coefficients c. 



64 TWO-DIMENSIONAL GEOMETRY 

3. Prove that the triangle formed by the diagonals of any complete 
quadrangle whose vertices are in the conic is a self-polar triangle. 

4. Prove that the triangle whose vertices are the diagonal points 
of a complete quadrilateral circumscribed about a conic is a self-polar 
triangle. 

5. Prove that a range of points on any line is projective with the 
pencil of lines formed by the polars of the points with respect to any conic. 

6. If P 19 P 2 , P 3 are three points on a conic, prove that the lines Pc i P l 
and P 2 P 3 are harmonic conjugates with respect to the tangent at P 2 and 
the line joining P 2 to the point of intersection of the tangents at P 1 and P 3 . 

7. If the sides of a triangle pass through three fixed points while 
two of the vertices describe fixed lines, prove that the locus of the third 
vertex is a conic. 

8. The equation ./J+ A/ 2 =0, where / x and / 2 are quadratic poly 
nomials and X is an arbitrary parameter, defines a pencil of conies. 
Sketch the appearance of the pencil according to the different ways 
in which the conies /j = and / 2 = intersect. 

9. Prove that through an arbitrary point goes one and only one 
conic of a given pencil and that two and only two conies of the pencil 
are tangent to an arbitrary line. What points and lines are exceptional ? 

10. Show that any straight line intersects a pencil of conies in a set 
of points in involution. What are the fixed points of the involution ? 

11. Prove that the polars of the same point with respect to the 
conies of a pencil form a pencil of lines. 

12. If the point P describes a straight line, prove that the vertex of 
its polar pencil (Ex. 11) with respect to the conies of a pencil describes 
a conic. 

13. Prove that the locus of the poles of a straight line with respect 
to the conies of a pencil is a conic. 

14. Prove that the conies of a pencil of conies which intersect in 
four distinct points have one and only one common self-polar triangle. 

15. Prove that the pole of the line at infinity is the center of the 
conic unless the conic is tangent to the line at infinity. 

16. Prove that the tangents to a central conic at the extremities of a 
diameter are parallel. 

17. Two lines are conjugate with respect to a conic if each passes 
through the pole of the other. Prove that each of two conjugate 
diameters is parallel to the tangents at the ends of the other. Prove 
also that a system of parallel chords are all conjugate to the same 
diameter and therefore bisected by it. 



CURVES OF SECOND OKDEK AND SECOND CLASS 65 

18. Consider a pencil of lines with its vertex at the center of a conic, 
and an involution in the pencil such that corresponding lines in the 
involution are conjugate diameters of the conic. Show that the fixed 
lines of the involution are the asymptotes. 

19. The foci are defined as the finite intersections of the tangents 
from the circle points at infinity to any conic. Show that a real central 
conic has four foci, two real and two imaginary, and that the real foci 
are those considered in elementary analytic geometry. 

35. Classification of curves of second order. We are now ready 
to find the simplest forms into which the equation 



can be put by a change of coordinates. 
As before let us place 

11 12 13 



CASE I. D = 0. The curve has no singular points ( 33), and 
there can be found an infinite number of self-polar triangles 
(VII, 34). Let one such triangle be taken as the triangle of 
reference. Then, since the polar of 0:0:1 is the line x s = 0, we 
shall have, in the new equation of the curve, # 13 = ^ 23 0. Since the 
polar of :1: is x z = 0, we shall have 12 = a^= 0. Since the polar 
of 1 : : is x l = 0, we shall have # 12 = a lg = 0. The equation of the 
curve is therefore ^ + ^ + ^ = Q _ (2) 



No one of the coefficients a n , 22 , a ss can be zero, for if it were 
the curve would have a singular point. 

If the coordinates of the original equation of the curve are real 
and the new coordinates are referred to a real self-polar triangle 
with a real unit point, the coefficients a n , # 22 , and # 33 are real. We 
may then distinguish two cases according as all or two of the signs 
in (2) are alike. By replacing "v/Ja^ja^ by x t we have then two 
types of equations, ^ + ^ + ^ = Q> (3) 

af + ^-a^O. (4) 

The first equation represents a curve with no real points and 
the other represents one which has real points. It is obvious that 
no real substitution can reduce one equation to the other. Of 



66 TWO-DIMENSIONAL GEOMETRY 

course the second equation can be reduced to the first by placing 
x s = ix^ which does not involve imaginary axes but an imaginary 
value of the constant k s . Summing up, we have the theorem : 

A curve of second order ivhose equation has real coefficients and 
which has no singular point is one of two types : an imaginary curve 
the equation of which can be reduced to the form (3), and a real curve 
the equation of which can be reduced to the form (4). If no account 
is taken of imaginaries the equation of any curve of the second order 
with no singular point can be reduced to the form (3). 

CASE II. D = 0, but not all first minors of D are zero. The 
curve has then one and only one singular point ( 33). This may 
be taken as the point 0:0:1. Then a 13 = a^ a 8S 0. The points 
0:1:0 and 1:0:0 may be taken in an infinite number of ways so 
that each is on the polar of the other. Each of these polars passes 
through 0:0:1 (IV, 34). Since 0:1:0 is the pole of x 2 = we 
have a lz = in addition to a^ 0, as already found, which is also 
the condition that 1:0:0 is the pole of x 1 = 0. The equation of 
the curve is therefore 



^\ 
. (5) 

Neither of the coefficients a n or 22 can be zero, for if it were, 
the curve would have more than one singular point. 

Equation (5) may be reduced without the use of imaginary 
quantities to one of the types 

^ + ^ = 0, (6) 

f r *?-0. (7) 

Summing up, we have the theorem: 

A curve of the second order whose equation has real coefficients and 
which has one singular point is one of two types : two imaginary straight 
lines represented by equation (6) or two real straight lines represented 
by equation (7). If no account is taken of imaginaries a curve of 
second order with one singular point consists of two straight lines inter 
secting in that point, and its equation may be put in the form (6). 

CASE III. D 0, and all its first minors are zero. The curve has 
then a line of singular points, and its equation may be reduced to 
#* = ( 33). A curve of second order with a line of singular points 
consists of that line taken double. 



CURVES OF SECOND ORDER AND SECOND CLASS 67 

EXERCISES 

1. Apply the foregoing discussion to the classification of curves in 
Cartesian coordinates, using x 3 = as the equation of the line at infinity. 
Where does the parabola occur in the discussion ? (See Ex. 2, 34.) 

2. Show from the foregoing that if an ellipse or a hyperbola is 

x 2 if 
referred to a pair of conjugate diameters, its equation is j^ = 1, 

and conversely. 

3. Show from the foregoing that if a parabola is referred to a diam 
eter * and a tangent at the end of the diameter, the equation of the 
parabola is y 2 = ax, and conversely. 

4. Show that if a central conic does not pass through either of the 
circle points at infinity, it has one and only one pair of conjugate 
diameters which are orthogonal to each other. 

5. Show that if a parabola does not pass through a circle point 
at infinity one and only one pair of axes described in Ex. 4 will be 
orthogonal. Write the equation of a parabola tangent to the line at 
infinity in a circle point. 

36. Singular lines of a curve of second class. Consider the curve 
of second class defined by the equation in line coordinates 

24^=0. (A ti = A ik ) (1) 

By 32 the singular lines of this locus are defined by the 
equations 



Let A, called the discriminant of the curve (1), be defined by 
the equation 

A = 

There are then three cases in the discussion of equations (2). 

CASE I. A = 0. Equations (2) have no solution, and the curve 
has no singular line. This is the general case. 

*A diameter of a parabola is defined as a straight line through the point of 
tangency of the parabola with the line at infinity. 



68 TWO-DIMENSIONAL GEOMETKY 

CASE II. A= 0, but not all the first minors of A are zero. 
Equations (2) have one solution, and the curve has one singular 
line. Let this line by a change of coordinates be taken as the line 
0:0:1. The degree of the equation will not be changed, but in 
the new equation we shall have A u = A 2S = A 83 = 0. The equation 

therefore becomes A 

A u vJ + 2A l . 2 u l u. 2 +A 22 u 2 = 0, 

which can be factored into two linear factors. These factors can 
not be equal, for if they were we should have A u : A 12 =-A 12 : A^ and 
equations (2), written for the new equation, would have more than 
one solution. Each of the factors of (3) represents a pencil of 
lines the vertex of which lies on the line x & = ; that is, on the 
singular line of the locus of (1). Equation (1) is the line equation 
of the two vertices of the pencils represented, and the singular line 
is the line connecting these two vertices. 

CASE III. A= 0, and all its first minors are zero. Any solution 
of one of the equations (2) is a solution of the others, and the 
curve has a pencil of singular lines. If by a change of coordinates 
that pencil is taken as the pencil u^ 0, we shall have in the new 
equation (1) A^=A IS =A 22 = A 2S = A 8S = 0, and the equation becomes 
M* = 0. Hence in this case equation (1) is the equation of two 
coincident points. 

Summing up, we have the following theorem: A curve of the 
second class has in general no singular line. If it has one singular 
line it consists of two distinct points lying on that line. If it has a 
pencil of singular lines it consists of the vertex of that pencil doubly 
reckoned. 

37. Classification of curves of second class. By 32 the limit 
points of intersection of two lines of the locus 

2)4t,.= (4*= 40 (1) 

are given by the equations 



There are again three cases corresponding to the cases of the 
previous section. 



CURVES OF SECOND ORDER AND SECOND CLASS 69 

CASE I. A = 0. Equations (2) can be solved for u^ u 2 , and u s , 
and the results substituted in (1). But by aid of equations (2), 
equation (1) can be replaced by the equation 



The result of the substitution is therefore 



X \ A \\ A 11 A IS 

X 2 A U A -22 A K 



= 0, 



(4) 



1 2 8 

which may be written ^, a ik x i x k~ ^> 

where a ik is the cofactor of A ik in the expansion of the determi 
nant A. 

This is the curve of second class enveloped by the lines which 
satisfy equation (1). It appears that it is also a curve of second 
order. Let 

Al <*12 ^13 



be the discriminant of (5). Then 

AGO 



D- A = 



A 
A 



-A 8 



and 



We have therefore the following result: A curve of second class 
with no singular line is also a curve of second order with no singular 
point. The converse theorem is easily proved : A curve of second 
order with no singular point is also a curve of second class with no 
singular line. 

Since the simplest equations of the curve of second order are 



the simplest equations of the curve of second class are 



70 TWO-DIMENSIONAL GEOMETRY 

CASE II. A= 0, but not all its first minors are zero. Equations 
(2) have no solution, so that no point equation can be found for 
the locus of the limit points on the lines of equation (1). In fact, 
we have already seen that the limit points are two in number 
only, the vertices of the two pencils of lines defined by (1). The 
simplest forms into which equation (1) can be put without the 
use of imaginary coordinates are obviously 



CASE III. A= 0, and all first minors are equal to zero. We have 
already seen that the simplest form of the equation in this case is 

uf = 0. 

38. Poles and polars with respect to a curve of second class. 
Equations (2), 37, can be used to establish a relation between 
any line u., whether or not it satisfies (1), 37, and a point x i de 
fined by these equations. The point is called the pole of the line, 
and the line is called the polar of the point with respect to the 
curve of second class given by equation (1), 37. The following 
theorem is then obvious : 

To any line of the plane corresponds a distinct pole, but to any 
point corresponds a distinct polar when and only when the discrim 
inant of the curve of second class does not vanish. 

This relation is dualistic to that of 34, and all theorems of that 
section can be read with a change of " point " to " line," " pole " to 
" polar," etc. We shall prove in fact that in case of a curve of second 
order and second class without singular point or line the definitions of 
poles and polars in 34 and 38 coincide. 

This follows from the fact that the curve of second class defined by 

^A ik u,u k = 
is, when A = 0, the curve of second order 



where a ik is the cof actor of A ik in A. Now, if equations (2), 37, 
are solved for u^ u z , and u z , there result the equations (1), 34, and 
the theorem is proved. 




CURVES OF SECOND ORDER AND SECOND CLASS 71 

In case a curve of second class consists of two points, by a 
theorem dualistic to IV, 34, the pole of any line lies in the 
singular line, which is the line connecting the two points. It may 
be found by means of a theorem which is dualistic to VIII, 34, and 
which may be worded as follows : 

If any point M is taken on a 
line p, and r and s are the lines 
through M belonging to a curve 
of second class, and q is the line 
joining M to the pole of p, the 
lines p and q are harmonic con 
jugates with respect to r and s. FIG. 20 

This theorem is illustrated in Fig. 20, which also suggests 
the construction necessary to find P the pole of jt?, since P is the 
intersection of q and the line 00 . 

EXERCISES 

1. If the three vertices of a triangle move on three fixed lines and 
two of its sides pass through fixed points, the third side will envelop 
a conic. 

2. A range of conies is defined by the equation yj + X/ 2 = 0, where 
f^ = and f 2 = are the equations in line coordinates of two conies. 
Discuss the appearance of the range. 

3. Prove that there is in general one and only one conic of a range 
which is tangent to a given line and two and only two conies of a 
range which pass through a given point. What are the exceptional lines 
and points ? 

4. Prove that for a given range all tangents through a fixed point 
form a pencil in involution with itself. 

5. Prove that for a given range of conies the poles of a fixed straight 
line form a range of points. 

6. If a straight line in Ex. 5 turns about a point, show that the base 
of the range of its polar points envelop a conic. 

7. Prove that the centers of the conies of a range lie on a straight 
line. 

8. Prove that the conies of a range with four distinct common 
tangents have one and only one self-polar triangle. 



72 TWO-DIMENSIONAL GEOMETRY 

39. Projective properties of conies. We shall prove the following 
theorems which are connected with the curves of second order and 
involve projective pencils or ranges. 

/. The points of intersection of corresponding lines of two projective 
pencils which do not have a common vertex generate a curve of second 
order which passes through the vertices of the pencils. 

Without loss of generality we may take the vertices of the two 
projective pencils as ^4(0 : : 1) and (7(1 : : 0) (Fig. 21) respec 
tively, and may take the point of intersection of one pair of 
corresponding lines as J5(0 : 1 : 0). The 
two pencils are then 

t^+** B 
and x 2 +\ f x s = 0, 

where X = - % The point B lies on 

7 A, + o A 

the line of the first pencil, for which 
X = 0, and on the line of the second 
pencil, for which X = oc. Since these are 

V V Xl_ A. --L FlG 21 

corresponding lines in the projectivity, 

we have B = 0. Then /3 and y cannot vanish, owing to the condition 

a8 (3y = 0. Now, if x l : x z : x s is a point on two corresponding lines 

of the pencils, we have X = -- -> X = -- 2 , and hence 




The point x-.x : x therefore lies on a curve of second order. 

i 123 

Conversely, if y l : y z : y z is a point on this curve of second order, 

-a+fi* 

v y 

we have = - 

y* y 

But the line joining y. to A has the parameter X = and 

y 

the line joining y i to B has the parameter X = , and conse 
quently X = - Hence the point y { is the intersection of 

two corresponding lines of the two projective pencils. 

That the curve of second order with the equation (1) passes 
through A and C is obvious. Hence the theorem is proved. 



CURVES OF SECOND ORDER AND SECOND CLASS 73 

If a = the curve (1) reduces to the two straight lines x 2 = 
and 72^ /3^ 3 = 0, and the two pencils are in perspective ( 16). 
Equation (1) may be written in the more symmetrical form 



or 



(2) 



II. The lines connecting corresponding points of two projective 
ranges which do not have the same base envelop a curve of second 
class which is tangent to the bases of the two ranges. 

This is dualistic to I. We may take the bases of the two ranges 
as a(0 : : 1) and c(l : 0: 0) (Fig. 22) respectively, and a line 
connecting two pairs of corresponding points as 5(0:1: 0). The 
line equations of points on the 
two ranges are then 

and w 2 -h\ w 3 =0, 
where, as for I, 

X/= *XHK 



The lines connecting corre 
sponding points then satisfy 
an equation of the form 



or 



u 




FIG. 22 



Conversely, any line satisfying this equation is a line connecting 
corresponding points of the two ranges. 

When a = the equation factors into u^ and 7^ /3u 8 = 0, 
and the two ranges are in perspective. 

///. Any two points on a curve of second order without singular 
lines may be used as the vertices of two generating pencils. 

No three points of the curve lie in a straight line. Hence any 
three points on the curve may be taken as the vertices of the 
coordinate triangle ABC. The equation of the curve is then of 
the form ^ r 0.^^^ -*- cxx =0, (4) 



where c^ c a , e g are not zero, since the curve has no singular point. 



74 TWO-DIMENSIONAL GEOMETRY 

The equation of any line through A is x l + \x 2 = and that of 
any line through C is x z + V# 8 = 0. If these lines intersect on (4) 
we have g.X-^ 

c s \ 

The correspondence of lines of the pencil with vertex A and 
those of the pencil with vertex C is therefore projective. This 
proves the theorem. 

IV. Any two tangent lines to a curve of second class without singular 
points may be taken as the bases of two projective generating ranges. 

This is dualistic to theorem III. 

V. If any point of a curve of second order without singular points 
is connected with any four points on the curve, the cross ratio of the 
four connecting lines is constant for the curve. If any tangent line to 
a curve of second class without singular lines is intersected by any 
four tangents, the cross ratio of the four points of intersection is 
constant for the curve. 

This is a corollary to theorems III and IV. 

VI. One and only one curve of second order can be passed through 
five points, no four of which lie in a straight line. 

Let the five points be %, P z , P A , P^ and P 5 (Fig. 23). 

From PI, which cannot be in the same straight line with P^P*, and 
P, draw the lines Pf^, P^P Z , P^P ; and from P 5 , which also cannot be 
collinear with P z , P z , P, draw P b P z , P 5 P 3 , P^. 
Then there exists one and only one pro- 
jectivity (I, 13) between the pencil with 
vertex JJ and that with vertex P b in which 
the line J?P Z corresponds to Jj?JJ, the line 
%% to P,P S , and the line P^ -to JJ.JJ. The 

intersection of corresponding lines of these 

FIG. 23 
projective pencils determine a curve ol 

second order through the five given points. Since any two points 
on the curve may be taken as the vertices of the generating pencils, 
only one curve can be passed through the points. 

VII. One and only one curve of second class can be constructed 
tangent to five lines no four of which meet in a point. 

This is dualistic to theorem VI. 




CURVES OF SECOND ORDER AND SECOND CLASS 75 



VIII. Pascal s theorem. If a hexagon is inscribed in a curve of 
second order, the points of intersection of opposite sides lie on a 
straight line. 

By a hexagon is meant in this theorem the straight-line 
figure formed by connecting in order the six points P^ P v P z , P 4 , 
JJ, jP, taken anywhere on the curve of second order (Fig. 24). 
The opposite sides are then P^ and 

pp ~P~P anrl PP PP iinrl PP 

JT. 1 , -^i dllLl _r-g, JT^JL. dllU. -*8-M 

respectively. 

We shall first assume that the curve 
is without singular points. Then the 
points .ZJ, J^, and P 6 do not lie on a 
straight line and may be taken as the 
vertices of the triangle of reference. 
Let P 1 be the point (0:0:1), P 3 
the point (0:1: 0), and P 5 the point p \ 
(1:0: 0). Then the equation of the 
curve is, by (2), 

r + - + T = 0- (5) 




Let P% have the coordinates $ 
P the coordinates 2 f , and J^ the 
coordinates w { . Then, since the three points P,, 
on the curve (4), we have 

L JL 1 

y\ y* y* 
III 

111 

W, W^ IV, 



FIG. 24 



and - lie 



= 0. 



(6) 



The equation of the line P^ is y^ y^ z = and that of Pf b 
is z 3 x^ z 2 xf oThey intersect in the point :!: Similarly, the 






lines PE and PP intersect in the point : - : 1 and the lines 



76 



TWO-DIMENSIONAL GEOMETRY 



P.P. and R P. intersect in the point !:-*:-* The condition that 

w t i 
these three points lie on a straight line is 



= 0, 



which is readily seen to be the same as equation (6). 

If the curve of second order consists of two intersecting straight 
lines, the theorem is still true, but the proof needs modification. 
When the points ^, J, and ^ lie on one of the straight lines 
and P^, P, % lie on the other, we have the theorem of Pappus 
(VII, 30). Other distributions of the points on the straight 
lines are trivial. 

IX. Brianchorfs theorem. If a hexagon is circumscribed about a curve 
of second class, the lines connecting opposite vertices meet in a point. 

This is dualistic to VIII, and the proof is left to the student. 



EXERCISES 

1. Prove that the center of homology (see Ex. 9, 30) of two pro- 
jective pencils of lines is the intersection of the tangents at the vertices 
of the pencils to the conic generated by the pencils. 

2. Prove that the axis of homology (see Ex. 10, 30) of two pro- 
jective ranges is the line joining the points of contact of the bases of 
the ranges with the conic generated by the ranges. 

3. Show that the lines drawn through a fixed point intersect a conic 
in a set of points in involution, the fixed points of the involution being 
the points of contact of the tangents from the fixed point. 

4. Prove that if two triangles are inscribed in the same conic they 
are circumscribed about another conic, and conversely. 

5. Prove that if a pentagon is inscribed in a conic the intersections 
of two pairs of nonadjacent sides and the intersection of the fifth side 
and the tangent at the opposite vertex lie on a straight line. 

6. State and prove the dualistic theorem to Ex. 5. 



CURVES OF SECOND ORDER AND SECOND CLASS 77 

7. Prove that if a quadrilateral is inscribed in a conic the inter 
sections of the opposite sides and of the tangents at the opposite 
vertices lie on a straight line. 

8. State and prove the dualistic theorem to Ex. 6. 

9. If a quadrilateral A BCD is inscribed in a conic and L is the 
intersection of the tangent at A and the side BC, K is the intersection 
of the tangent at B and the side AD, and M is the intersection of the 
sides AB and CD, prove that Z, K, and M lie on a straight line. 

10. State and prove the dualistic theorem to Ex. 8. 

11. If a triangle is inscribed in a conic, prove that the intersections of 
the tangents at the vertices with the opposite sides lie on a straight line. 

12. State and prove the dualistic theorem to Ex. 12. 

13. Prove that the complete quadrangle formed by four points of 
a conic has, as diagonal points, the points of intersection of the 
diagonal lines of the complete quadrilateral formed by the tangents 
at the vertices of the complete quadrangle. 



CHAPTER VI 

LINEAR TRANSFORMATIONS 

40. Collineations. A collineation in a plane is a point trans 
formation ( 5) expressed by the equations 



a 12 x 2 + 

px 2 = a 2l x L + a 22 x 2 + V 3 , (1) 

px 3 = (1^+ a S2 x 2 + a S3 x s . 

If the determinant \a ik \ is not equal to zero, these equations can 
be solved for x { , with the result 

ax l = A u x( 4- A 2l x 2 + A sl x 3 , 

ax 2 = A 12 x[ + A 22 x f 2 + A S2 x 3 , (2) 



where J tt is the cof actor of a ik in the expansion of | a lk and where 

Ml*o. 

If the determinant |a iA; |=0, equations (2) cannot be obtained 
from (1). For this reason it is necessary to divide collineations 
into two classes: 

1. Nonsingular collineations, for which |%|^0. 

2. Singular collineations, for which |# ijfc |=0. 

We shall consider only nonsingular collineations in this text, 
though some examples of singular collineations will be found in 
the exercises. 

It is obvious that for a nonsingular collineation x { cannot have 
such values in (1) that x[ = x 2 x s = 0. Hence by (1) any point x i 
is transformed into a unique point x[. Similarly, from (2) any 
point x\ is the transformed point of a unique point x t . 

Consider now a straight line with the equation 



78 



LINEAR TRANSFORMATIONS 79 

All points #,., which satisfy this equation, will be transformed into 
points x\, which satisfy the equation 

where, by (2), 



It appears then that any straight line with coordinates u t is 
transformed by (1) into a unique line with coordinates u\. Also, 
equations (3) may be solved for u { with the result 

Xw 1= a u u[ 4- a 2 X + 8 X 

Xtt 2 = 12 Wj + 2 X + 32 W 3 (4) 

Xw 3 = a l3 u[ 4- aX 4- gX 

from which it appears that any line is the transformed line of a 
unique line. 

Equations (3) express in line coordinates the same transforma 
tion that is expressed by equations (1) in point coordinates. For 
it is easy to see that by equations (3) any pencil of lines with the 
vertex x { is transformed into a pencil of lines with the vertex x\ and 
that the relation between x { and x\ is exactly that given by equa 
tions (1). Equations (3), therefore, which express a transformation 
of straight lines into straight lines, also afford a transformation of 
points into points in a sense dualistic to that in which equations (1) 
afford a transformation of straight lines into straight lines. 

We will sum up the results thus far obtained in the following 
theorem : 

I. By a nomingular collineation in a plane every point is trans 
formed into a unique point and every straight line into a unique 
straight line and, conversely, every point is the transformed point of a 
unique point and every straight line the transformed line of a unique 
straight line. 

Consider now a collineation R l by which any point x i is trans 
formed into the point x(, where 



and let R 2 be a collineation by which any point x\ is transformed 
into xl\ where ,, 





80 TWO-DIMENSIONAL GEOMETRY 

Then the product R Z R 1 is a substitution of the form 



which is a collineation. Hence the product of two collineations is 
a collineation. 

Moreover, if R 1 is as above and R 2 is of the form 

the product RM. is .. .. 

2 TX = X, TX = X a , TX = X~. 

1 I - 2 " ** * 

which is the identical substitution. Hence in this case R Z is the 
inverse substitution to R l and is denoted by R^ 1 . Our work shows 
that the inverse transformation to a collineation always exists and 
is itself a collineation. 

These considerations prove the following theorem : 

//. The totality of nonsingular collineations in a plane form a group. 

We shall now prove the following theorems : 

///. If II, P 2 , PS, P are any four arbitrarily assumed points, no three 
of which are on the same straight line, and l, P 2 , P 3 r , P are also 
four arbitrarily assumed points, no three of which lie on a straight 
line, there exists one and only one collineation by means of which jFJ is 
transformed into P^ , P 2 into Pi, P s into P%, and P into P^. 

To prove this we will first show that one and only one collinea 
tion exists which transforms the four fundamental points of the 
coordinate system, namely ^(0 : : 1), J5(0 : 1 : 0), (7(1 : : 0), and 
/ (1 : 1 : 1), respectively, into four arbitrary points P^ (o^ : a 2 : # 3 ), 
P 2 (ft : ft : ft), P a ( 7l : 7a : 7s ), and P, (8, : S 2 : S 8 ), no three of which 
lie on a straight line. 

By substituting in equation (1) the coordinates of correspond 
ing points, remembering that the factor p may have different values 
for different pairs of points, we have the following equations out 
of which to determine the coefficients a ik : 



LINEAR TRANSFORMATIONS 81 

By substitution from equations (5) in equations (6) we have 



which may be solved for pj p 2 : p s : p^. Since no three of the points 
JJ, P 2 , J^, 7J lie on a straight line no determinant of the third 
order formed from the matrix 




can vanish, and hence no one of the factors p. can be zero. The 
values of p^ p z , p s , and p 4 having thus been determined except for 
a constant factor, the values of the coefficients a ik can be found 
from (5) except for this same factor. Hence the collineation (1) 
is uniquely determined, since only the ratios of a ik in (1) are 
essential. 

Let it now be required to transform the four points P, P^ JJ, P 4 , 
no three of which are on a straight line, into the four points P, 
1%, PJ, PJ, respectively, no three of which are on a straight line. 
As we have seen, there is a unique collineation E l which transforms 
A, B, C, I into jf, P 2 , J^, JJ respectively, and a unique collinea 
tion R z which transforms A, B, C, I into If, P 2 , JJ , P 4 r respectively. 
Then the collineation j^" 1 (theorem II) exists and transforms 
P 1 , P 2 , P%, PI into A, B, C, I respectively. The product R. 2 R l is 
a collineation (theorem II) which transforms JJ, P 2 , P 8 , P into 
PI, Pj, P^, PJ respectively. Moreover, this is the only collineation 
which makes the desired transformation. For let R be a collinea 
tion which does so. Then R~ 1 R transforms Jj% ZJ, P s , P into 
A, B, C, I respectively. Hence 



whence R = R 2 Ri l . 

This establishes the theorem. It is not necessary that all the 
points 7^, P 2 , P z , P 4 should be distinct from the points PJ, PJ, Pj, P. 
In the special case in which P^ is the same as 7J ; , Jo tne same as 






82 TWO-DIMENSIONAL GEOMETKY 

PJ, P z the same as JJ , and P the same as P, R l = R 2 and R is 
the identical substitution. Hence we have as a corollary to the 
above theorem: 

IV. Any collineation with four fixed points no three of which are in 
the same straight line is the identical substitution. 

V. Any nonsingular collineation establishes a projectivity between 
the points of two corresponding ranges and the lines of two correspond 
ing pencils, and any such projectivity may be established in an infinite 
number of ways by a nonsingular collineation. 

To prove the first part of the theorem let the point y { be trans 
formed into y\ and the point z f be transformed into z t by the collinea- 
tion (1), so that 



Then y { -f- \z { is transformed into f t , where 

ftfi = a i i G/i + **i) + i 2 O* + ** 2 ) + ,- 3 Q/3 + ^3) 

= />!*/! +VA ; 

whence o-f i = ?/J + A/^ 

where X/=^- 

Pi 

This establishes a projectivity between the points of the range 
y i + \z i and those of the range y\ -f- X gJ . By the use of line coordinates 
and equations (3) the proof may be repeated for the lines of a pencil. 

To prove that there are an infinite number of nonsingular col- 
lineations which establish a given projectivity between the points 
of two ranges, it is only necessary to show that there are an infinite 
number of collineations which transform any three points P, $, R 
lying on a straight line into any three points P\ Q\ R , also on a 
straight line, and apply III, 15. 

To prove this, draw through R any straight line and take S and 
T two points on it. Draw also through R any straight line and 
take S and T any two points on it. 

Then by theorem III there exists a collineation which trans 
forms the four points P, Q, S, T into the four points P , # , S , T , 
and this collineation transforms R into R . Since S, T and S r , T 1 
are to a large extent arbitrary, there are an infinite number of 
required collineations. 



LINEAR TRANSFORMATIONS 



83 



If it is required to determine a collineation which establishes a 
projectivity between two given pencils of lines, this may be done 
by establishing a projectivity between two ranges, each of which 
is in perspective with one of the pencils. Since this may be done 
in an infinite number of ways, there are an infinite number of the 
required collineations. 

41. Types of nonsingular collineations. A collineation has a, fixed 
point when #/=#,- in equations (1), 40. The fixed points are 
therefore given by the equations 



. - /O X z 



CO 



The necessary and sufficient conditions that these equations have 
a solution is that p should satisfy the equation 



= 0. 



(2) 



Similarly, the fixed lines of the collineation are given by the 
equations - - 



and the necessary and sufficient condition that these equations 
have a solution is 



= 0. 



Equations (2) and (4) are the same and will be written 






(4) 



= 0. (5) 

Now let p l be a root of (5). Then p l cannot be zero, since by 
hypothesis | a ik \ = 0. The root p l is a double root when 

n-Pi a i* _ a n-Pi a u = (6) 
a o a a 

31 33 "\ 2 1 22 ri 

and it is a triple root when 

/"O,)= 2 [(-/,)+ (-/>,)+ (-/,)]= 0. (7) 



: 



84 TWO-DIMENSIONAL GEOMETRY 

We may now distinguish three cases : 

1. When all the first minors of the determinant f(jp^) do not vanish. 
Equations (1) and (3) have each a single solution. The collineation 
has then a single"!ixecl. point and a singly fixed line corresponding to 
the value p r The root /> may be a simple, a double, or a triple root 
of (5), according as equations (6) and (7) are or are not satisfied. 

2. When all the first minors of f(jp^) vanish, but not all the 
second minors vanish. Equations (1) and (3) contain then a single 
independent equation. The collineation has then a line of fixed 
points and a pencil of fixed lines corresponding to the value p^ 

The root p l is at least a double root of (5) since equation (6) is 
necessarily satisfied, and it may or may not be a triple root. 

3. When all the second minors of f(jp^) vanish. Equations (1) 
and (3) are satisfied ,by all values of x i anjl u { respectively, and 
the collineation leaves all points and Twines fixed. The root p 1 is then 
a triple root of (5) since equations (6) and (7) are satisfied. 

From this it follows that a collineation has as^many fixed lines as 
fixed points and as many pencils of fixed lines as lines of fixed points. 

From 12 it follows also that in every fixed line lies at least one 
fixed point and that through every fixed point goes at least one fixed 
line. The line connecting two fixed points is fixed and the point common 
to fixed lines is fixed. 

We are now prepared to classify collineations according to their 
fixed points and to give the simplest form to which the equations 
of each type may be reduced. We will first notice, however, that 
if the point x. 0, x s = 0, x k =\ is fixed, then by (1), 40, 
a ik = a jk = ; and if the line x k = is fixed, then a ki = a kj = 0. 

A. Collineations with at least three fixed points not in the same 
straight line. Take the fixed points as the vertices A, B, C of the 
triangle of reference. Then the collineation is 

px( = a n x r 
px 2 = 



No one of the coefficients can be zero, since the collineation is 
nonsingular, but they may or may not be equal. We have then 
the following types, in writing which different letters are used to 
indicate quantities which are not equal. 



LINEAR TRANSFORMATIONS 85 

TYPE I. px( ax v 

p4 = 1v 
px f 3 = cx v 

The collineation has only the fixed points A, B, C and the 
fixed lines AB, BC, and CD. 

TYPE II. P x[ = ax,, 

P4 = ax 2 , 
px 8 = cx s . 

The collineation has the fixed point A, the line of fixed points 
BC, the fixed line BC, and the pencil of fixed lines with vertex A. 
It is called a homology. 

TYPE III. px[ = x 



All points and lines are fixed. It is the identical transformation. 

B. Collineations with at least two distinct fixed points, but no others 
not in the same straight line. We will take the two fixed points 
as A (0 : : 1) and C (1 : : 0) of the triangle of reference. The 
collineation has at least two distinct fixed lines one of which is AC. 
The other must contain one of the fixed points, and we will take 
it as BC (# 3 = 0). The collineation is then 



px! 2 = 



Here a 12 = or we should have case A. We shall place 12 =1. 
The equation (5) is now ( n /3)( 22 /3)( 33 />)= 0. Placing 
P ~ a w we h ave as ^ ne equations to determine the corresponding 
fixed point 



Since by hypothesis every fixed point lies on # 2 = 0, we have 
a n = a 22 * ^ ^ s l 6 ^ undetermined whether 83 is or is not equal to a^. 
Hence we have two new types. 



86 TWO-DIMENSIONAL GEOMETRY 

TYPE IV. px( = ax l + # 2 , 



px! 2 = 



The collineation has only the fixed points A and C and the 
fixed lines AC and BC. 

TYPE V. px( = ax l + x. 2 , 

px[ 2 = ax. 2 , 



The collineation has the line of fixed points AC and the pencil 
of fixed lines with its vertex at C. 

In either Type IV or V the point B may be taken at pleasure 
on the line BC. 

C. Collineations with only one fixed point. Take the fixed point as 
C (1:0:0). The collineation has also a fixed line which must 
pass through C. Take it as BC (# 3 = 0). The collineation is now 



Equation (5) is now ( u /3)( 22 />) ( 33 / > )= 0, and since 
by hypotheses C is the only fixed point, we have a n = a 22 ^= 33 . 
The point A (0:0:1) taken at pleasure is transformed into 
A ( 13 : a zs : 33 ), and if we take the line AA f as x l = 0, we have 
13 = 0. The coefficients 12 and a 23 cannot vanish or we have the 

/> 

previous cases. We may accordingly replace x z by 2- and x ^ by 

x 12 

^ and have, finally, 

iA 8 

TYPE VI.* px( = ax l + x v 

px 2 = a 



*The above classification has been made by means of geometric properties. 
The reader who is familiar with modern algebra should compare the classifica 
tion by means of Weierstrass s elementary divisors. Cf . Bocher s " Higher Algebra," 
p. 292. 



LINEAR TRANSFORMATIONS 87 

EXERCISES 

1. Find the fixed points and determine the type of collineation to 
which each of the following transformations in Cartesian coordinates 
belong : (a) a translation, (&) a rotation about a fixed point, (c) a reflection 
on a straight line. 

2. Determine the group of collineations in Cartesian coordinates 
which leaves the pair of straight lines x 2 y^= invariant and discuss 
the subgroups. 

3. Are two collineations with the same fixed points always commu 
tative ? Answer for each type. 

4. Consider the singular collineations. Prove that there is always a 
point or a line of points for which the transformed point is indeter 
minate. We shall call this the singular point or line. If there is a 
singular point, every other point is transformed into a point on a fixed 
line which may or may not pass through the singular point. If there 
is a singular line, every point not on the line is transformed into a 
fixed point which may or may not lie on the singular line. Prove these 
facts and from them show that the singular collineations consist of the 
following types : 

I. One singular point P, a fixed line p not through P, two fixed 
points on p. , 



II. One singular point P, a fixed line not through P, one fixed 



III. One singular point P, a singular line p not through P, all points 
of p fixed. , 



= x. 



IV. One singular point P, a fixed line p through P, one point oip fixed. 

px[ = x s) 

0t*/2 U./2 j 



88 TWO-DIMENSIONAL GEOMETRY 

V. One singular point P, a fixed line^? through P, no point of p fixed. 

px[ = x 2 , 
px 2 = x a , 

,4=0. 

VI. A singular line p, a fixed point P on p. 

px[ = x s , 
P x 2 = 0, 
P ^=0. 

VII. A singular line p, a fixed point P not on p. 



42. Correlations. The equations 

2 x 2 + a ls x s , 

0) 



pu(= a 



where x i are point coordinates and u are line coordinates, define 
a transformation of a point into a line. Such a transformation is 
called a correlation. As in the case of collineations, we shall dis 
tinguish between nonsingular and singular correlations according 
as the determinant | % | does not or does vanish, and shall consider 
only nonsingular correlations. Equations (1) can then be solved for 
x, with the result 



where A ik is the cof actor of a ik in the determinant | a ik \. Every 
straight line w/ is therefore the transformed element of a point x-. 
Consider now the points of a line given by the equation 



where u. are constants. By (2) these points go into a pencil of 
lines the vertex of which is the point #/, where 



(3) 



LINEAR TRANSFORMATIONS 89 

We may express this by saying that the line u< is transformed 
into the point x . Also, since equations (3) can be solved for u. 
with the result 



every point is the transformed element of one and only one line. 
Since equations (2), (3), and (4) are consequences of equations 
(1), we shall consider them as given with (1) and sum up our 
results in the following theorem: 

7. A nonsingular correlation defined by equations (1) is a trans 
formation by which each point is transformed into a straight line and 
each straight line into a point, in such a manner that points which lie on 
a straight line are transformed into straight lines which pass^ through a 
point, and lines which pass through a point are transformed into points 
which lie on a straight line. Each line or point is transformed into one 
point or line and is the transformed element of one line or point. 

Consider now a correlation S l by which a point x { is transformed 
into a line u!, and let S be a correlation by which the line u( is 
transformed into a point x". It is clear that the product S^ is a 
linear transformation by which the point x i is transformed into the 
point x" ; that is, a collineation. Therefore the correlations do not 
form a group. It is evident, however, that the inverse transformation 
of any correlation exists and is a correlation. 

We can therefore prove the following theorems : 

77. If Jf, P z , P z , P are four arbitrary points, no three of which lie 
on a straight line, and if p^ p z ,p z , P 4 are four arbitrary lines, no three 
of which pass through a point, there exists one and only one correla 
tion by means of which P l is transformed into p^, P 2 into p z , P s into p s , 
and P into p 4 , and there exists also one and only one correlation by 
means of which p r is transformed into JFJ, p 2 into P 2 , p s into P%, and 
/(into P 4 . 

III. Any nonsingular collineation establishes a projectivity between 
te points of a range and the lines of a corresponding pencil, and any 

ch projectivity may be established in an infinite number of ways by 
a correlation. 



90 TWO-DIMENSIONAL GEOMETRY 

The proofs of these theorems are the same as those of the cor 
responding theorems of 40 and need not be repeated. 

By equations (1) a point x { lies on the line u!, into which it is 
transformed when and only when 



+ (<**,+ a 2 )*A=- (5) 

That is, allies on a conic K^ 

Similarly, from equations (3) a line u. passes through the point 
x i9 into which it is transformed when and only when 



,= o. (6) 

That is, u. envelops a conic K^. 

It is evident that the conies K^ and K Z are not in general the 
same. Their exact relations to each other will be determined later 
in this section. In the meantime we state the above result in 
the following theorem : 

IV. In the case of any nonsingular correlation the points which lie 
on their transformed lines are points of a certain conic, and the lines 
which pass through their transformed points envelop a certain conic, 
which, in general, is not the same as the first. 

Any point P of the plane may be considered in a twofold manner : 
as either an original point which is transformed by the correlation 
into a line or as a transformed point obtained from an original 
line. If P is an original point it corresponds to a line p 1 whose 
coordinates are given by (1). If P is a transformed point it corre 
sponds to a line p whose coordinates are given by (4), in which we 
must replace x[ by ar f , the coordinates of P. 

The lines p and p do not in general coincide. When they do 
the line p and the point P are called a double pair of the correlation. 
That P should be a point of a double pair it is necessary and suffi 
cient that the coordinates u[ and w t . of equations (1) and (4) should 
be proportional; that is, that the coordinates of P should sa^ ^fy 
the equations ^ 

On - P a J x i + 12 - P a J x * + Ois - P a J \ = 



LINEAR TRANSFORMATIONS 91 

where p is an unknown factor. For these equations to have a solu 
tion it is necessary and sufficient that p should satisfy the equation 

* u -P*ii 

= 0. (8) 

The correlations may be classified into types according to the 
nature of the double pairs and of the conies K^ and K^. As a pre 
liminary step we shall prove the theorem : 

V. If the point P and the line p form a double pair, then p is the 
polar of P with respect to the conic K^. 

To prove this let the coordinates of P be y^ where y t is the solution 
of (7) for p p^ and let v i be the coordinates of p. Then v i is deter 
mined from (1) when x { is replaced by y { . Then from (1) and (7) we 
have x- i , N . 

whence 

/ \ 1 \ 

v pJ v 

These last equations are exactly those which determine the polar 
of P with respect to K^ and the theorem is proved. 

We now proceed to the classification. 

A. Let K l be a nondegenerate conic. By a proper choice of coordi 
nates its equation can be put in the form 

so that a n = a^= 0, 81 = 18 a 82 = ~ a ^ a \z^ ~~ a zi 

If there is at least one double pair of which the point is not on the 
conic, it may be taken as A (0 : : 1) without changing the form of 
equation (9). We shall then have 18 = 23 = 0. The correlation is 
now expressed by the equations 



pui = a n : 



Neither a w nor 21 can be zero. There are then two types accord 
ing as 12 and 21 are or are not equal : 

TYPE I. pu( = ax 21 

K = ax v 



92 



TWO-DIMENSIONAL GEOMETRY 



The conic K l has now the equation # 3 2 -f-2 ax^ 2 = Q^ and the correla 
tion is a polarity with respect to this conic. Conversely, any polarity 
with respect to a nondegenerate conic can be expressed in this form. 

The equation (8) now becomes 2 (1 p)*= 0, and equations (7) 
are identically satisfied when p = 1. Hence in a polarity every cor 
related point and line form a double pair. The equation (6) now 
becomes au+ %UjU 2 = 0, which is the line equation of K. Henc 
in a polarity the conies K^ and K^ coincide. 

TYPE II. pu[ = ax z , 

) pu , = bx^ 



The conic 7f 2 has the line equation 



or the point equation 



and the relation of the two conies K^ and K z is as in Fig. 25. Equa 
tion (8) becomes (\_ \ / _ j WA _ ^ 

which has three unequal roots. The correlation has accordingly 
three double pairs : namely, the point A and the line BC, the point 
B and the line AB, the point 
C and the line AC. 

Types I and II arise from 
the assumption that there is 
a double pair of which the 
point lies outside the conic. 
If there is no such pair, there 
must be at least one of which 
the point lies on the conic. 
In this case take the point as J, IG 

B (0 : 1 : 0) without changing 

the form of equation (9). By theorem V the line of the double 
pair which contains B is the tangent BA. Then, from (1), a^= 0. 
We have before seen that # 23 = 82 , so that the correlation is now 

f . 

pu l = 12 3^-|- # 13 # 3 , 





LINEAR TRANSFORMATIONS 



93 



The coefficient # 13 cannot be zero or we should have the previous 
case. The equation (8) is now ( 12 pa zl ) ( 21 /^ 12 ) (1 p) = 0, 
and the solution p = 1 would give a point not on K^ contrary to 
hypothesis, unless 21 = la . We have, finally, for the equations of 
the correlation: 



TYPE III. 



K = 
K = 



pu( = bx l + x# 
where a = b is not excluded. The line equation of JT 2 is now 

5 2 w 2 2 cful 2 au^ = 0, 
and the corresponding point equation is 



The two conies K l and 7f 2 lie therefore in the position of Fig. 26. 

The equation (8) for p has the triple root /> = !, and the cor 
relation has only one double pair consisting of the line point B 
and the line AB. 

B. Let the conic K^ degenerate 
into two intersecting straight 
lines. We may take the equa 
tions of the lines in the form 

whence 




FlG 



The point B is again taken 
as the point of a double pair 

and is therefore transformed into a line through B, and if we 
take that line as 2^= we have, from (1), a 32 = 0. The equation (8) 



where 12 cannot be zero since the correlation is nonsingular. 
The root p = 1 gives the point B as a point of a double pair. 
The root p = 1 gives the point : a u : a 12 , and if this be taken as 
A we have #,= 0. 



94 



TWO-DIMENSIONAL GEOMETRY 



We have then, finally, 
TYPE IV. pu[ = 



bx 



pu = * 

where the equality of the coefficients is not excluded. 
The conic K Z has now the equation 

aul 4- b 2 u 2 = 0, 

which is that of two pencils with their vertices on AB. The relation 
of K I and jfif 2 is shown in Fig. 27. 

C. Let the conic K^ degenerate into two 
coincident straight lines. Take the equa- A 

tion of K l as x i _ Q^ 

The discussion proceeds as in the pre 
vious case with the coefficient a placed 
equal to zero. We have, accordingly, 



TYPE V. 



= - bx 



2 , 



pu 2 = 




FIG. 27 



The conic K 2 has the equation u* = 0, which is that of a double 
pencil of lines with the vertex A. The relation of the two conies 
JTj and JT 2 is shown in Fig. 28. The equation (8) now becomes 



The root p = 1 gives the point A as 
a point of a double pair of which the 
line is BC. The root p = 1 gives 
any point on the line J5(7, so that if M 
is any point on BC it is a point of a ~~ 
double pair the line of which is AM. / 

EXERCISES 




FIG. 28 



1. Find the square of each of the different types of correlations and 
determine the type of collineation to which it belongs. 

2. Prove that if P is a point on K l the two tangents drawn from P 
to K 2 are the two lines to which P corresponds in the correlation 
according as P is considered as an original point or a transformed point. 



LINEAR TRANSFORMATIONS 95 

3. Prove that if p is a tangent to K 2 the two points in which p inter 
sects K^ are the two points to which p corresponds in the correlation 
according as p is considered as an original line or a transformed line. 

4. Take any point P. Show that the line into which P is transformed 
by a correlation of Types II, III, V is a line which connects two of the 
four points of intersection with K l of the two tangents drawn from P 
to K 2 . Show also that the line which is transformed into P is another line 
connecting the same four points of intersection. Determine these two 
lines more exactly and explain the construction in Type IV. 

5. Take any line p. Show that the point into which p is transformed 
by a correlation of Types II, III, V is one of the four points of inter 
section of the four tangents drawn to K 2 from the points in which p 
intersects A^. Show also that another of these points of intersection is 
the point which is transformed into p. Determine these points more 
exactly and explain the construction in Type IV. 

6. Show that if every point lies in the line into which it is trans 
formed by a correlation, the correlation is a singular one of the form 



-h a 



Study the correlation. 

43. Pairs of conies. The preceding results may be given an 
interesting application in studying the relation of two conies to 
each other, especially with reference to points and lines which are 
the poles and polars of each other with respect to both the conies. 

Let 



and ]W*=0 (2) 

be two conies without singular points. The product of a polarity 
with respect to (1) and a polarity with respect to (2) is a non- 
singular collineation which may be expressed by the equations 



(3) 



The fixed points of the collineation (3) are identical %ith the 
points which have the same polars with respect to both (1) and (2), 
and the fixed lines of (3) are identical with the lines which have the 



96 



TWO-DIMENSIONAL GEOMETRY 



same poles with respect to (1) and (2). Each fixed point of (3) 
will be paired with some fixed line of (3) as pole and polar. These 
points and lines we shall refer to briefly as common polar elements. 
We shall have as many arrangements of common polar elements 
as there are arrangements of fixed points of (3) and may classify 
them into the types given in 41. 

TYPE I. There are three and only three common poles A, B, C 
(Fig. 29) and three common polars AB, BC, CA. To pair these off 
we notice first that no point can be the pole of a line through it. 

For if B were the pole of 
AB, for example, C would be 
the pole of either AC or BC, 
say AC. The lines AB and 
A C would be tangent to each 
of the conies (1) and (2) and 
A would be the pole of BC. 
Then if D were any point 
whatever on BC, and E its 
harmonic conjugate with re 
spect to B and C, the line 
EA would be the polar of 
D with respect to both (1) 
and (2). Hence the conies would have more than three common 
polars, and the collineation (3) would not be of Type I, 41. 

Therefore the triangle is a self-polar triangle with respect to 
both (1) and (2). By taking this triangle as the coordinate tri 
angle, the equations of the conies reduce to the forms 

af + ^+a^O, (4) 

arf+a^+asX^O, (5) 

and the collineation (3) becomes 




FIG. 29 



px{=a l x l , 



(6) 



where, by 41, a^ a^ a s . 

The two conies (4) and (5) intersect in four distinct points, as 
is easily proved. 



LINEAK TRANSFORMATIONS 



97 



TYPE II. There are two common poles A and C (Fig. 30) 
and two common polars AC and BC. The point C must be the 
pole of one of the lines AC and BC 
which pass through it, and hence 
C lies on the two conies. But C 
cannot be the pole of BC, for, if 
it were, A would be the pole of 
AC, and the line AC would be tan 
gent to the conies at A and in 
tersecting them again at C, which 
is impossible. Therefore C is the 
pole of AC and A of BC. If we 
take the axes of coordinates as in 

Type IV, 41, the equation of each of the conies is of the form 
a^+a^+Za^x^O. (7) 

Without changing the position of the axes we may take one of 
the conies as x * + ^ + 2 ^ = Q, (8) 

leaving the equation of the other in the general form (7). The 
collineation (3) is then , _ 




c 

FIG. 30 



or 



rt = 

p4 = 



(9) 



That this should be of Type IV, 41, we must have ^ = 8 , a 2 i= a & . 

The conies (1) and (2) are tangent at C and intersect in two other 
points, as is easily proved. The 
conies have no common self-polar 
triangle since there are not three 
fixed points in the collineation (9). 

TYPE III. There is a line BC 
(Fig. 31) each point of which is 
a common pole and another com 
mon pole A not on BC. The FlG 31 
common polars consist of the line BC and all lines through A. It 
is evident that A is the common pole of BC, and hence BC is not 




98 



TWO-DIMENSIONAL GEOMETRY 



tangent to the conies. Take as B any point of BC and take C as the 
pole of AB. Then ABC is a common self -polar triangle. The equa 
tions of the two conies may now be written as in Type I, (4) and (5), 
with the addition that now a^= a z , in order that the collineation (6) 
should be of Type II, 41. Hence the equations of the conies are 
reduced to the forms 2 2 2 A ._, AN 

0?+2?+3JBQ| (10) 

^+^+^=0, (11) 

and the collineation (3) becomes 



(12) 



The two conies are tangent at two points, namely the points in 
which the line BC meets the conies. This is easily seen from the 
equations. We may also argue that if BC meets (10) in L, the 
point L is a common pole of the line AL. Hence AL is tangent 
to both conies. Similarly, if M is the other point of intersection 
of BC and (10), AM is a common tangent to the conies. 

TYPE IV. There is one common pole C (Fig. 32) and one com 
mon polar BC. Hence the two conies are tangent to BC at C 
and tangent at no other point. Take any point on the conic (1) as 
A, and the tangent to (1) at A as AB. 
The equation of (1) then is 
x+ 2x^=0, 

while that of (2), since it is known 
only to be tangent to BC at (7, is 

ajX% 4- a<pl + 2 a s x.,x s + 2 a 4 x^ = 0. 
The collineation (3) is then of the type 




32 



px{= (1^+ a z x 2 

In order that this should have 

i ,., 

only one fixed point it is necessary 

and sufficient that a l = a^ a s 3= 0. The two conies, besides being 
tangent at (7, intersect in the point x l :x. 2 :x 8 = a} : 4 8 a a : 8 3 ". 



LINEAR TRANSFORMATIONS 



99 



If this point is taken as the point A in the coordinate triangle, 
we have 2 = 0. The equations of the conies are then 

x* 2 + 2x^=0, (13) 

22^=0, (14) 



and the collineation (3) is 
px[ =%!+ 
px( = x. 2 + 



= ax 



ax. 



(1 5) 



which is of Type VI, 41. 

As noted, the two conies are tangent at one point and intersect 
in another point. 

TYPE V. There is a line BC (Fig. 33) of common poles and a 
pencil, with vertex C on BC, of common polars. Every point on BC 
is therefore the common pole of some line through C, and hence 
C is the common pole of BC. Hence the two conies are tangent to 
BC at C. We proceed as in Type IV, but we 
now find that in order that all points on # 3 = 
should be fixed points of the collineation we 
must have a^ = a^ a z 0. The equations of the 
conies therefore reduce to 



and the collineation (3) becomes 
px, = x, H- ax.) 



= x n 




(18) 



FIG. 



which is of Type V, 41. 

The two conies are tangent at one point and have no other point 
of intersection. 

TYPE VI. Every point of the plane is a common pole with 
respect to the two conies. The two conies are obviously identical. 

To each type of the arrangements of the common polar elements 
corresponds a distinct kind of intersection of the two conies. 
Conversely, the nature of the common polar elements is deter 
mined by the nature of the intersections, as is easily proved. 



100 TWO-DIMENSIONAL GEOMETRY 

It is sometimes important to find, if possible, a self-polar triangle 
common to two conies. The foregoing discussion leads to the 
following theorem: 

If two conies intersect in four distinct points they have one and 
only one common self -polar triangle. If they are tangent in two points 
they have an infinite number of common self -polar triangles, one vertex 
of which is at the intersection of the common tangents. In all other 
cases two distinct conies have no common self-polar triangle. 

It is only when two conies have a common self -polar triangle 
that their equations can be reduced each to the sum of squares 
as in Types I and III. 

EXERCISES 

1. Prove that the diagonal triangle of a complete quadrangle whose 
vertices are on a conic, or of a complete quadrilateral whose sides are 
tangent to a conic, is self-polar with respect to the conic ; and, con 
versely, every self-polar triangle is the diagonal triangle of such a quad 
rangle and such a quadrilateral. Corresponding to a given self-polar 
triangle one vertex or one side of such a quadrangle or such a quadrilat 
eral may be chosen arbitrarily. Apply this theorem to determining the 
common self-polar triangle of two conies in the position of Type I. 

2. Discuss the common polar elements of a pair of conies when one 
of them has singular points, obtaining seven types corresponding to 
the seven types of singular collineations given in Ex. 4, 41. (Notice 
that if the conic (1) consists of two intersecting straight lines, the point 
of intersection P is the singular point of the corresponding collineation, 
and the polar p of P with respect to the conic (2) is the fixed line. If the 
conic (1) consists of a straight line taken double, that line is the singular 
line p, and its pole P with respect to the conic (2) is the fixed point.) 

44. The protective group. As we have seen, the product of two 
collineations is a collineation, and the product of two correlations 
is a collineation. It is not difficult to show that the product of a 
collineation and a correlation in either order is a correlation. The 
inverse transformation of either a collineation or a correlation 
always exists and is a collineation or a correlation respectively. 
Hence we have the theorem : 

The totality of nonsingular collineations and nonsingular correla 
tions in a plane form a group, of which the collineations form a 
subgroup. 



LINEAR TRANSFORMATIONS \ 101 

This group is called the protective group, and protective geometry 
consists of the study of properties which are invariant under this 
group. 

It is evident then that projective geometry will include the study 
of straight-line figures with reference to the manner in which lines 
intersect in points or points lie on straight lines. Such theorems 
have been illustrated in 30. Lengths of lines are not in general 
invariant under the projective group, and projective geometry is 
not therefore concerned with the metrical properties of figures. 
The cross ratio of four elements is, however, an invariant of the 
projective group, and hence the cross ratio is of importance in 
projective geometry. 

By means of a collineation any conic without singular points 
may be transformed into the conic 



This was virtually proved in 35 when we showed that any equa 
tion of the second order with discriminant not zero may be reduced 
to the above form. But any transformation of coordinates is ex 
pressed by a linear substitution of the variables, and this substitution 
may be interpreted as a collineation, the coordinate system being 
unchanged. Hence any conic without singular points can be trans 
formed into any other conic without singular points by a collineation. 
Similarly, any conic with one singular point may be transformed 
into any other conic with one singular point, and any conic with 
an infinite number of singular points may be transformed into any 
other which also has an infinite number of singular points. Hence 
projective geometry recognizes only three types of conies and studies 
the properties which are common to all conies which belong -to each 
of the types. Such properties are illustrated in the theorems of 
39, where the distinction between ellipse, hyperbola, and parabola 
is not made. 

In projective geometry it is convenient sometimes to consider the 
properties invariant under the subgroup of collineations. The corre 
lations may be implicitly employed by use of the dualistic property. 

45. The metrical group. We shall proceed to study the collinea 
tions which leave all distance invariant or multiply all distances 
by the same constant k. For that purpose it is convenient to use 



: S/lt I 

< <T *. n 

1-02 TWO-DIMENSIONAL GEOMETRY 

Cartesian coordinates. Since it is evident that all points at infinity 
remain at infinity, the transformations must be of the form 

px r = a^ + a$ + a s t, 

j + bfr (1) 



or in nonhomogeneous form 

J=ax + a + a sJ 



Transformations of this type are called affine, since any point 
in the finite part of the plane is transformed into a similar point. 
We proceed to find the conditions under which an affine transfor 
mation will have the properties required above. 

If (x^ y^) and (x 2 , y^) are any two points which are transformed 
respectively into (#(, ?/[) and (#, ^) then, by hypothesis, 

& - x y+ <y, _ y y = 

from which we obtain 



Since this must be true for all values of the variables, we have 

oj +!?-, 



a^a. 2 -\- bfi 2 = 0. 

From this follows algebraically 6 2 =a 1 , 5 1= = T a 2 . Also an 
angle can always be found such that a^=~k cos <, ^ = k sin 0. 
Equations (2) can then be written 

x =k(x cos </> ^/ sin </>) + a, 

(3) 
y = lc (x sin < + ?/ cos 



The product of any two transformations of the form (3) is 
also of the form (3). This can be shown by direct substitution, 
or follows geometrically, since (3) is the most general collineation 
which multiplies distances by a constant. It is also evident that 



LINEAR TRANSFORMATIONS 103 

the inverse transformation of (3) exists and is of the same form. 
Hence the following theorem: 

7. Transformations of the form (3)/orm a group called the metrical 
group of collineations. 

To this we add the following theorem : 

77. By the metrical group of collineations the circle points at infinity 
are either fixed or interchanged with each other. Conversely, any col- 
lineation which leaves the circle points fixed or interchanges them 
belongs to the metrical group. 

This follows from the fact that minimum lines (19) must be 
transformed into minimum lines. Since the line at infinity is fixed, 
the points wr^re the minimum lines intersect the line at infinity 
must be fixea or interchanged. Theorem II may therefore be 
restated as follows: ^ 

777. The metrical group leaves invariant the curve of second class 
consisting of the two circle mi&ts at infinity!* 

We shall now enumerate certain special types of the trans 
formation (3). 

I. Translation. 



This is of Type V, 41, the line of fixed points being the line 
at infinity, and the pencil of fixed lines being the parallel lines 
intersecting in a : b : 0. 

The translations evidently form a subgroup of the metrical 
group. 



II. Hotation about a fixed point. 

If the fixed point is the origin, we have the transformation 

= x cos </> y sin </>, 
y 1 = x sin < -f- y cos <j>. 



This is of Type I, 41, the fixed points being the origin and the 
two circle points at infinity. 



104 TWO-DIMENSIONAL GEOMETRY 

A rotation about any other point is the transform ( 5) of R by T. 
Thus, if R is a rotation about (a, 6), R = TRT~\ where E is the 

transformation , , , , , N . , 

r 2; a = (x a) cos </> (j/ 6) sin <, 

{ y 6 = (# a) sin < + (y b) cos c/>. 

The substitutions R and 72 form each a subgroup of the 
metrical group. 

III. Magnification. 

( x 1 = kx, 
M 



This is of Type II, 41, the fixed point being the origin, and 
the line of fixed points being the line at infinity. The pencil of 
fixed lines is the pencil with its vertex at (0, 0). 

A magnification M 1 with the fixed point (a, 5) is the transform 
of M by T ; thus, M f = TMT~ \ where M is the transformation 

f x a = k (x a), 
M 1y- b = k(y- J). 

The transformations M and M form each a subgroup of the 
metrical group. 

IV. Reflection on a straight line. 

If the straight line is the axis of x, the transformation is 



s \y =-y- 

This is of Type II, 41, the line of fixed points being y = 0, 
and the distinct fixed point being 0:1:0. The fixed pencil of lines 
consists of the parallel lines through 0:1: 0. 

If, now, U is a transformation of the metrical group (3), it is not 
difficult to show that it is the product of transformations of the 
types we have enumerated. There are, in fact, two main divisions 
of the metrical transformations, namely, 

CLASS I. Metrical transformations not involving a reflection. 
Consider U^ = TMR. It is evident that U^ is given by the equations 
( x = k (x cos (f> y sin <) -f- a, 

and that, conversely, any transformation of this type can be ex 
pressed as the product TMR. 



LINEAR TRANSFORMATIONS 105 

CLASS II. Metrical transformations involving a reflection. 
Consider U 2 = TSMR. It is evident that V is of the type 

r x = k(x cos (/> y sin $) + a, 
2 1 y = k (x sin < + / cos </>) + 6, 

which can also be written 



( x = k (x cos (j> + y sin <) + a, 
J^ s (2 sin < ?/cos<) + 6 

by replacing by </>, an allowable change, since </> is any angle. 

Conversely, any transformation of type U 2 can be expressed as 
the product TSMR. 

The transformations U 1 form a subgroup of the metrical group. 
The transformations Z7 2 , however, do not form a group, since the 
product of two such transformations is one of the form U^ 

46. Angle and the circle points at infinity. By the metrical group 
angles are left unchanged. This is evident from the fact that any 
triangle is transformed into a similar triangle. Also the cross ratio 
of any two lines and the minimum lines through their point of inter 
section is equal to the cross ratio of the transformed lines and the 
minimum lines through the transformed point of intersection, since 
minimum lines are transformed into minimum lines. This suggests 
a connection between this cross ratio and the angle between the 
two lines. We shall proceed to find this connection. 

Let the two lines be ^ with line coordinates v f , and l z with line 
coordinates w { . The coordinates of any line through the point of 
intersection of ^ and l z are u { = v { + \w^ and this is a minimum line 
when u. satisfies the line equation of the circle points at infinity, 

namely, 

u *+ u *=0. 

This gives for X the equation 

A\*+ZB\+ (7=0, 
where A=wf + w, B w^v^+v^w^ C = v* + v. 




106 TWO-DIMENSIONAL GEOMETRY 

and call m l the minimum line corresponding to \ 15 and m z the 
minimum line corresponding to \ 2 . Then (13) 

... , \ -B + i^/AC-B* 

== 



Now the point equations of ^ and 1 2 are respectively 

y + v f = > 
y + = 0, 



and if </> is the angle between them, 

v,w, 4- vjtv* B 



sin </> = 



VAC 

X, cos <f> i sin 6 e*** 
- 1 - 



,-, 
Therefore 

cos <> =F ^ sin 



whence <^> = - log - 1 . 

J \ 

The ambiguity of sign is natural, since an interchange of \ and 
X 2 would change the sign of </>. We have, therefore, 



7 

n^Ze between two lines is therefore equal to times the 

,& 

logarithm of the cross ratio of the two lines and the minimum lines 
through their point of intersection. 

If = ~, i = 1, and, conversely, if -* = -!, <f> = ^ + far. 

TT ^ ^2 ** * 

Hence 

Perpendicular lines may be defined as lines which are harmonic 
conjugates with respect to the minimum lines through their point of 
intersection. 



CHAPTER VII 

PROJECTIVE MEASUREMENT 

47. General principles. The results of the last section suggest a 
generalization, to be made by replacing the circle points at infinity 
by the general curve of the second class, 

"^k ~A A f\ S J 4 \ /"I \ 

which we shall call the fundamental conic. Let ^ and / 2 (Fig. 34) 
be any two lines, and let ^ and t z be the two tangents which can be 
drawn to the fundamental conic from the 
point of intersection of ^ and 1 2 . Then the 
projective angle between ^ and 1 2 is defined 
by the equation 

4(^ 2 ) = M Iog(y 2 , ^ 2 ), (2) 

where M is a constant to be determined 
more exactly later. 

This satisfies the fundamental require 
ments for the measurement of an angle, 
since it attaches to every angle a definite 
numerical measure such that the sum of the measures of the parts 
of a whole is equal to the measure of the whole. To prove the 
latter statement, notice that 




Now, if Z 1? l z , and l s are three lines of the same pencil, with coor 
dinates \ t , \ 2 , \ respectively, and the coordinates of the lines ^ 
and t z of the same pencil are taken as and oo, we have 



Hence 



107 



108 



TWO-DIMENSIONAL GEOMETRY 



Dualistically, if the fundamental conic does not reduce to two 
points its equation can be expressed in point coordinates as 

% = 0. (OK = a ik ) (3) 



Then, if ^ and F 2 (Fig. 35) are two points, and T^ and T 2 are 
the two points in which the line I^P 2 cuts the conic, the protective 
distance P^ is defined by the equation 

dist.(^) = Jnog(^, I\T 2 ), (4) 

where K is to be determined later. It is 
shown, as in the case of angles, that 
dist. (JJJ?) + dist. (JSJP) = dist. (J?^). 

The analytic expression for distance ft, 
and angle in terms of the coordinates of 
the points and lines, respectively, may 
readily be found. Take, for example, 

equation (4). If y i are the coordinates of ^, and z. the coordinates 
of P 2 , the coordinates of T v and T 2 are y. \z t and ^- \z., where 
\ and X 2 are the roots of the quadratic equation 




FlG 



which we write for convenience in the form 



We will take 



X =^B 



and 



g) "- 

Then, by the definition (^TJ and theorem III, 13, we have 

dist. (j/ t -2 ) = K log 
\ 



But 






and therefore we have, as the final form, 
dist. (j/f2 f ) = 2 K log * 



PROJECTIVE MEASUREMENT 109 

There is of course free choice as to which of the two values of 
X is taken as \. To interchange \ and \ is simply to change the 
positive direction on the line. 

The distance between two points is zero when the two points 
are coincident or when the line connecting them is tangent to the 
fundamental conic, since in the latter case \= \ 2 . The tangents to 
the conic are therefore analogous in the projective measurement 
to the minimum lines in ordinary measurement. 

The distance between two points is infinite when \ x or X 2 is 
zero or infinity. This happens only when ^ or P^ is on the funda 
mental conic. That is, points on the fundamental conic are at an 
infinite distance from all other points. 

Similarly, consider equation (2). If v { and w { are the coordinates 
of Z x and 1 2 respectively, the coordinates of ^ and t z are v. \^w { and 
v i \w { , where \ and \ 2 are the roots of the equation 



which may be written 

n -2xn 



vw 



we take 



we have, by (2), 






4 (,,0 = If log = 2 M log n.^ . (7) 



All angle is zero if ^ and 1 2 coincide or if ^ and 1 2 intersect on the 
fundamental conic, for in the latter case \= \. That is, all lines which 
intersect at an infinite distance make a zero angle with each other. They 
are therefore analogous to parallel lines in Euclidean measurement. 

The angle between two lines is infinite if either line is tangent 
to the fundamental conic. 

From the definitions we have the following theorem : 

Projective distance and angle are unchanged by the group of collin- 
eations which leave the fundamental conic invariant. 

We shall now proceed to discuss more in detail three cases, 
according to the nature of the fundamental conic. 



110 



TWO-DIMENSIONAL GEOMETKY 



48. The hyperbolic case. We assume that the fundamental conic 
is real. It may then be brought by proper choice of coordinate 
axes to the form 

in point coordinates and to the form 

in line coordinates. 

The conic divides the plane into two portions, one of which 
we call the inside of the conic and which is characterized by the 
fact that the tangents to the curve from 
any point of the region are imaginary. 
The outside of the conic is the region 
characterized by the fact that from every 
point of it two real tangents can be 
drawn. We shall consider the inside of 
the conic. 

If l^ and 1 2 (Fig. 36) are two real 
lines intersecting in a point inside the 
conic, \ and \ of equation (7), 47, 
are conjugate imaginary. Let us place 
\ i = re** t where 




FIG. 36 




(2) 



Then \=re- { * and 

(lJ^) = Mloge* i * = 

Since it is desirable that the angles which a line makes with 
another should differ by multiples of TT, we shall place M- -> 

and have, as the complete definition of the angle between the 
lines Zj and Z 2 , Q _ 



mr 



whence 



cos 6 



(3) 



Two lines are perpendicular to each other when 6 =(2 
For that it is necessary and sufficient that = 1. The two lines 



PROJECTIVE MEASUREMENT 111 

are then harmonic conjugates with respect to t l and t 2 . This has a 
geometric meaning, as follows : Let P (Fig. 36) be the point of in 
tersection of l t and 2 , p the polar of P, L I and L 2 the intersections 
of p with ^ and 1 2 respectively, and T^ and T Z the intersections of 
the conic with ^ and t 2 respectively. T^ T 2 , t^ 2 , being imaginary, 
are not shown in the figure. Then by VI, 34, T I and T 2 lie on p, 
and by I, 16, (L^L^ ^T^) = (^ 2 > *A) Hence, in order that the 
two lines ^ and / 2 should be perpendicular it is necessary and suffi 
cient that L I and L 2 should be harmonic conjugates to T^ and T 2 , 
and hence (VIII, 34) L l must lie on the polar of L 2 , and L 2 
must lie on the polar of L^ But the polars of L I and L 2 pass 
through P by V, 34, and therefore ^ is the polar of Z 2 , and 2 
is the polar of L^. Hence for two lines to be perpendicular it is 
necessary and sufficient that each should pass through the pole of 
the other. 

Consider now the distance between two points P l and P 2 (Fig. 36) 
inside the conic. Then \ x and \ 2 of (5), 47, are both real, and 
hence if the distance PP is to be real we must take K as a real 



k 
quantity. Let us place JT=- where k is real. We have, for the 

distance, ~ 

dist. <,, O = % = k log 



If we write d for dist. Q/ t . t .) we have, from (4), 



.-^p^-v <-<%*.. 



whence 




We have already noted that if P^ is inside the conic and ^ on 
the conic, the distance P^ becomes infinite. If P l is inside the conic 
and j outside of it, X 1 and X 2 in equation (4) have opposite signs, 



112 



TWO-DIMENSIONAL GEOMETRY 



and the distance Pf^ becomes imaginary. If, then, we can imagine 
a being living inside the conic and measuring distance and angle by 
the formulas (5) and (3), the conic would lie for him at an infinite 
distance, and the region outside would be simply nonexistent, a 
mere analytic conception in which a point means simply a pair 
of coordinate values. Such a being would have a non-Euclidean 
geometry of the type named Lobachevskian. 

We have, of course, based all our discussion on the assumption 
of the Euclidean axioms, and the inside of our fundamental conic is 
simply a portion of the Euclidean plane. It lies outside the scope 
of this book to show that by a choice of axioms, differing from 
those of Euclid only in the parallel axiom, it is possible to arrive 
at a geometry which for the entire plane has properties which are 
exactly those of the interior of our fundamental conic, with the 
projective measurement here defined. Such a discussion may be 
found in treatises on non-Euclidean geometry. The inside of the 
fundamental conic is a picture in the Euclidean plane of the non- 
Euclidean geometry. We shall proceed to notice some of the most 
striking properties. 

We first notice that if LK (Fig. 37) is a straight line and P 
a point not on it, there go through P two kinds of lines, those which 
intersect LK and those which do not. 
The latter lines are those which in the 
entire plane intersect LK in points 
outside the conic, but from the stand 
point of the interior of the conic they 
must be considered as not intersect 
ing LK. The two classes of lines, the 
intersecting and the nonintersecting, 
are separated from each other by two 
lines PL and PK, which intersect LK on the conic; that is, at 
infinity. These lines we call parallel lines, and say that through a 
point not on a straight line can be drawn two lines parallel to that 
straight line. 

The angle which a line parallel to LK through P makes with 
the perpendicular to LK is called the angle of parallelism, and is a 
function of the length of the perpendicular. To compute it, let 
us take LK as x^ = 0, the point P as y^ and the equation o f 




FIG. 37 



PROJECTIYE MEASUREMENT 113 

the conic as xt + x%x%= 0. The pole of LK is (1:0 : 0). The 
line PR is perpendicular to LK when it passes through the pole 
of LK. Its equation is therefore y z x z y^x^ = 0, and it intersects 

Hence, if p is the length of PR we have, from (5), 



k --- k 



The point K is the point (0:1:1), and the equation of PK is 
(y<i~ y^) x -C~ y^i^r y^* 0- Hence to find the angle between PK 
and P.K we have to place in (3) 



There results, with the aid of (6), 



It appears, then, that the angle is a function of p. We shall 
place, following Lobachevsky s notation* 



Our last equation then leads with little work to the final result : 



This result is independent of the fact that it has been obtained 
for the special line 2^= and the special form of the equation of 
the conic since no transformation of coordinates alters the projective 
angles or distances. 

If in formula (5) we consider y. as a fixed point C and replace 
z i by a variable point #,., at the same time holding the distance d 

constant, we have 

< + =0 (8) 

as the equation of the locus of a point at a constant distance 
from a fixed point. This locus is called a pseudo circle. From 
the form of (8) it is obvious that the pseudo circle is tangent to 



114 



TWO-DIMENSIONAL GEOMETRY 



the fundamental conic co xx = at the points in which the latter is 
cut by the polar <o yx = of the point y.. There are three cases: 

I. The point C lies inside the conic (Fig. 38). The pseudo 
circles with the center .y i are then closed curves intersecting the 
conic in imaginary points. 

II. The point C lies on the conic (Fig. 39), and the distance of 
each point from y i is infinite. The pseudo circles are tangent to the 





FIG. 38 



FIG. 39 



conic. They are the limiting cases of the pseudo circles of Case I 
when the center recedes to infinity and the radius becomes infinite, 
and are called in non-Euclidean geometry limit circles or horicycles. 
III. The point C is outside the conic (Fig. 40), and the radius 
is imaginary so that points of (8) lie inside the conic. The straight 
line (i> yx = is one of these pseudo circles, and the others are the 
loci of points equidistant 
from this line. To prove 
the latter statement draw 
any straight line through C. 
It intersects the polar of C 
at R and the pseudo circle 
in two points one of which 
is Q. Then CR and CQ are 
constant, and hence RQ is 
constant. In this geometry, 
then, the locus of points equally distant from a straight line is 
not a straight line, but a pseudo circle with imaginary center and 
imaginary radius. It is called a hypocycle. 




FIG. 40 



PROJECTIVE MEASUREMENT 115 

EXERCISES 

1. Consider angle and distance for points outside the fundamental 
conic, especially with reference to real and imaginary values. 

2. Construct a triangle all of whose angles are zero. 

3. Compute the angle between two lines of zero length and between 
any line and a line of zero length. 

4. Prove that the sum of the angles of a triangle is less than two 
right angles. 

49. The elliptic case. We assume that the fundamental conic is 
imaginary. It may be reduced by proper choice of coordinates to 
the form ^ = x * + x * + x * = Q (1) 

in point coordinates and to the form 

ft MM =< + < + < = (2) 

in line coordinates. 

Since the tangents from any point to the fundamental conic 
are imaginary, the problem of determination of angle is the same 
here as in the hyperbolic case, and we have 

(3) 



Any straight line connecting the two points JFJ and P 2 meets the 
conic in imaginary points, and if P l and JJ are real points, the 
quantities \ and X 2 in (5), 47, are conjugate imaginary. Hence, 
if the distance between two real points is to be real, we must take 

ik 

K as pure imaginary. We will place K= -> where k is real. 

Placing \ = re**, where 




and representing the distance (j/^) by c?, we may reduce formula 
(5), 47, to the form d 



Two real points are always at a finite distance from each other, 
since, as shown in 47, an infinite distance only results when one 
of the points is on the fundamental conic. 

Consider the change in d as z i moves along a straight line, y { 
being fixed. In the beginning of the motion, when z. coincides 



116 TWO-DIMENSIONAL GEOMETRY 

with y^ cos - = uy , and the sign of the radical must be taken 





so that cos - 1 and d = 0. As z { moves away from y. the signs 
k 

of the quantities on the right-hand side of equation (4) remain 
positive and d increases until z i reaches a point on the line o> yx 0, 
(Fig. 41), the polar of ?/ r Then 

cos - = and d= k. This is 

ri 

true of all lines through y { 
and for either direction on any 
such line. Hence the straight 
line a> yx = 0, which, by 48, 
is perpendicular to all lines 
through y^ is at a constant 

distance from y. in all \ 

2 FIG. 41 

directions. 

Consequently, if we start from y. and traverse a distance irk on 
any line through y { and in either direction, we return to y^ There 
are two cases of importance to be distinguished: 

CASE I. All straight lines may be considered of length jrk. 
The coordinates y { always refer, then, to a single point. All straight 
lines intersect in one and only one point, there are no parallel 
lines, and two lines always bound a portion of the plane. This is 
the Riemannian geometry. It may be visualized by drawing straight 
lines from a point outside the plane and considering each point of 
the plane as represented by one and only one of these lines. 

CASE II. All straight lines may be considered of length 2 irk. 
When we traverse the distance TrJc on a line from y. and return to 
y# we shall consider that we are on the opposite side of the plane 
and need to repeat the journey to return to our starting point. 
Any coordinates y# then, are the coordinates of two points lying 
on opposite sides of the plane. Two straight lines intersect in two 
points, there are no parallel lines, and two lines inclose two por 
tions of the plane. We call this spherical geometry, since it is exactly 
that on the surface of a sphere. It is also the geometry of the half- 
lines or rays drawn to the plane from a point outside of it. 



PROJECTIVE MEASUREMENT 117 

EXERCISES 

1. Construct a triangle all of whose angles are right angles. 

2. Prove that the sum of the angles of a triangle is greater than 
two right angles. 

50. The parabolic case. We may consider that the fundamental 
conic is one which contains singular points or singular lines. 
There are, then, the two possibilities of the point equation repre 
senting two straight lines or of the line equation representing 
two points. The former possibility has little interest, and we shall 
consider only the case in which the line equation represents two 
points. There are two cases to distinguish : 

CASE I. The two points are imaginary. We may take them as 
the two points ~L:i: 0, and the line equation of the fundamental 
conic is then n ^ = ^ + ^ 2= Q> (1) 

The formula for angle may be modified as in 48, with the 
result that 



The point equation of the fundamental conic does not exist and 
the distance formula (6), 47, cannot be immediately applied. 
We may proceed, however, by a method of limits. In place of (1) 
we will write Q _ = <+ <+ ^ 0> (3) 

which goes over into (1) when e = 0. The point equation cor 
responding to (3) is 

=&+ *f) + *t=0, (4) 

and from this we find, as in 48, 

d = i^ ^ lZs ~ ^i) 2 + 
* 



Since the quantity on the right hand of this equation is infini 
tesimal, we may replace sinh- by j and then pass to the limit, 

K, K 

as e = and k = oo in such a manner that Lim ikvl = 1. We have 



118 TWO-DIMENSIONAL GEOMETRY 

If we take # 3 =0 as the line at infinity, the points l:i:Q 
become the circle points, and the formula (2) for angle and (5) 
for distance become the usual Cartesian formulas. The geometry 
is Euclidean. We have this result: 

^Euclidean measurement is a special case of protective measurement. 

CASE II. The fundamental points are real. We may take them 
as 1 : 1 : 0. The line equation of the fundamental conic is then 

Q, m =u*-u*=0. (6) 

Since through every real point there go two lines of the pencils 
defined by (6), it is necessary to take the constant K of 47 as 
real if real lines are to make real angles with each other. We 
will take K=\ and find, by a discussion analogous to that used 
in 48 for finding d, 

U /J VW ~ 

COsh 



The formula for distance may be found as in Case I, with the 
result . - 

d = v O/A- 



If we take # 3 = as the line at infinity and use nonhomogeneous 
Cartesian coordinates, we have, for the distance between two points 

*, and * ,</, 2 



and for the angle between the two lines ax + ly + c = and 
a x + Vy + c = 0, 

aa -bb 



cosh 6 = 



Consider now any fixed point in the plane. For convenience let 
it be the origin 0. Through go two lines of the pencils defined by 
the fundamental conic; that is, two lines drawn to the fundamental 
points at infinity. The equations of these lines are x y = 
(Fig. 42). They divide the plane into two regions, which we may 
mark as shaded and unshaded. If a point (x,.y) lies in the unshaded 
region, 2^ ?/ 2 > 0; and if it lies in the shaded region, a^ y 2 < 0. 
Consequently, distances measured from are imaginary in the 



PROJECTIVE MEASUREMENT 



119 



shaded region and real in the unshaded region. The boundaries 
between the two regions are lines of length zero. The locus of 
points equidistant from are equilateral hyperbolas x*y*=k. 

A line ax + by = 0, passing 
through 0, is in the unshaded 
region if a 2 b 2 < and in the 
shaded region if c?W> 0. Hence 
an angle with its vertex at is 
real if both sides are in the shaded 
region or both sides in the un 
shaded region, and is imaginary 
if one side is in the shaded region 
and one side in the unshaded 
region. A line through which 
is not a line of zero length makes 
an infinite angle with each of the 
lines of zero length. The two lines of zero length make an inde 
terminate angle with each other. In this respect as in other ways 
they are analogous to the minimum lines in a Euclidean plane. 

These properties are of course the same at all points of the 
plane. They make a geometry which differs widely from the 
geometry of actual physical experience.* 

*This geometry has recently gained new interest because of its occurrence 
in the theory of relativity. Cf . Wilson and Lewis, " The Space-Time Manifold 
of Relativity," Proceedings of the American Academy of Arts and Sciences (1912), 
Vol. XL VIII, No, 11. 




I 
FIG. 42 



CHAPTER VIII 

CONTACT TRANSFORMATIONS IN THE PLANE 

51 . Point-point transformations. Consider now the transformation 
defined by the equations 



(1) 



where x 19 # 2 , x s and x( 9 x* 2J x 3 are point coordinates and/ 1? f^f s are 
homogeneous functions which are continuous and possess deriva 
tives and for which the Jacobian 



a/, 

t/ 3 



does not identically vanish. 

By the transformation (1) a point x. is transformed into one 
or more points #J, with possible exceptional points. Owing to the 
hypothesis as to the Jacobian, equations (1) can in general be 
solved for x t , and any point x 1 . is therefore the transformed point 
of one or more points x fl with possible exceptional points. 

Consider now a point M and its transformed point M . If there 
is more than one transformed point, we will fix our attention on 
one only. If M describes a curve c denned by the equations 



the point M describes a curve c , the equations of which may be 
found by substituting from (2) into (1). The direction of c at 
M is determined by x^ x 2 , x 3 and dx^ dx z , dx^ as shown in (4), 
31. The direction of c 1 at M is determined in the same manner 

120 



CONTACT TRANSFORMATIONS IN THE PLANE 121 

by x{, #2, #3 and dx(, dx%, dx z . These latter six quantities are 
determined by the former six, and hence the direction of c at a 
point M 1 is determined by the direction of c at M. From this 
follows the theorem 

If two curves c l and c^ are tangent at a point M, the transformed 
curves c( and c! 2 are tangent at the transformed point M f . 

For this reason the transformation (1) is called a contact 
transformation. 

If the transformation (1) is expressed in nonhomogeneous 
Cartesian coordinates, it becomes 



Now let p be the direction -^ of a curve traversed by the point 

dy 
(x, y) and let p be the direction -^ of the transformed curve. 

We have, evidently, 

^ 

dx 



Sx dy 
The three equations 

*=f l (.x, y), 



y), (3) 



p = 



dx 



a/. 



are called an enlarged point transformation. They bring into clear 
evidence that two curves with a common point and a common 
direction are transformed into two curves which have also a 
common point and a common direction. 

52. Quadric inversion. An example of a point-point transforma 
tion as defined by (1), 51, has already been met in the case of 
the collineations. 



122 TWO-DIMENSIONAL GEOMETEY 

As another example consider the transformation 



(1) 



These equations can be solved when neither x v x 21 nor x 8 are 
zero into the equivalent equations 



(2) 



The transformation establishes, therefore, a one-to-one relation 
between the points x { and the points x\ with the possible excep 
tion of points on the triangle of reference ABC. To examine these 
points let A be as usual the point : : 1, B the point 0:1:0, and 
C the point 1 : : 0, so that the equation of AB is rc 1 = 0, that of 
AC is # 2 =0, and that of BC is x s = 0. Then from (1) any point 
on the line AB is transformed into B, any point on the line AC is 
transformed into C, and any point on the line BC is transformed 
into A. The coordinates of either A, B, or C, if substituted in (1), 
give the indeterminate expression 0:0:0, but if we enlarge the 
definition of the transformation by assuming that (2) holds for all 
points, including those on AB, AC, and BC, it follows that B is 
transformed into the entire line AB, C is transformed into the 
entire line AC, and A is transformed into the entire line BC. 
Consider any straight line with the equation 

<*&+<*&+<*&=<>. 
It is transformed into the curve 



which is a conic through the points A, B, and C. In fact, the point 
in which the line meets AB is transformed into B, the point in 
which the line meets AC is transformed into (7, and the point in 
which the line meets BC is transformed into A. 

If the straight line passes through one of the points A, B, or (7, 
the conic into which it is transformed splits up into two straight 
lines, one of which is a side of the coordinate triangle and the 
other of which passes through the vertex opposite that side. In 



CONTACT TEANSFOEMATIONS IN THE PLANE 123 

particular, consider a line a^-h \x 2 = through A. The first two of 
equations (1) give x(+\x z = for all points except the point A\ 
that is, any point except A on a line through A gives a definite 
point on the same line. The point A, however, goes over into the 
entire line x 8 = 0. 

In a similar manner a conic is transformed into a curve of 
fourth order, which passes twice through each of the points ^4, 5, (7, 
since the conic cuts each of the lines AB, BC, CA in two points. 
If, however, the conic passes through one of the points A, B, (7, 
that point is transformed into a side of the coordinate triangle, 
and the curve of fourth order must consist of that side and a 
curve of third order. 

In particular, a conic through A but not through B or C is 
transformed into the line BC and a curve of third order through 
B and C. A nondegenerate conic through B and C and not through 
A is transformed into two lines AB and A C and a conic through B 
and (7, but not through A. Finally, a nondegenerate conic 
through the three points A, B, C is transformed into the three sides 
of the triangle of reference and a straight line not through its ver 
tices. These results may all be seen directly or verified analytically. 

By placing x i =x i in equations (1) the locus of fixed points of 
the transformation is found to be the conic 

x^-x^Q, 
which passes through B and C and is tangent to AB and AC. 

It is not difficult to show that each point P of the plane is trans 
formed into a point P 1 in which the line AP cuts the polar of P 
with respect to the fixed conic. 

This transformation is called a quadric inversion to distinguish 
it from the circular inversion, or simply inversion, discussed in the 
next section. 

EXERCISES 

1. Prove the statement in the text that the point P is transformed 
into the point in which AP cuts the polar of P with respect to the 
fixed conic. Hence show that P and P are harmonic conjugates to the 
points in which PP 1 cuts the conic. 

2. Prove that the cross ratio of four points on a straight line p is 
equal to the cross ratio of the corresponding four points on the conic 
into which p is transformed. 



124 TWO-DIMENSIONAL GEOMETRY 

3. Study the transformations 



/<}\ n ~J rf. ~, 

\*) P X l X 1 X $ 



= x?. 



(3) px[ = xf, 



P x 3 = x* - 0^3. 

53. Inversion. The transformation (1) of 52 has particular 
interest and importance when the points B and C are the circle 
points at infinity. We may then place x 3 = t, x 1 = x + iy, x^ x iy 
and, using Cartesian coordinates, write the transformation in the 

form 




(1) 
or, wliat is the same thing in nonhomogeneous form, 



(2) 



By this transformation a one-to-one relation is established 
between the points (x, y) and (V, / ), with the exceptions that the 
origin corresponds to the line at infinity, and conversely, and that 
each of the circle points at infinity corresponds to the minimum 
line joining it to the origin, and conversely. The circle x*-\-y z = 1 
is fixed. Any point of the fixed circle is transformed into a point 



CONTACT TRANSFORMATIONS IN THE PLANE 125 

inside that circle, and, conversely, in such a way that if is the 
origin, P any point, and P 1 the transformed point, OP OP f = 1. 
The transformation is called an inversion with respect to the unit 
circle, or a transformation by reciprocal radius with respect to 
that circle. The origin is called the center of inversion, and the 
fixed circle the circle of inversion. 

Remembering that a circle is a conic through the circle points 
and applying the results of the previous section, we have the 
following theorems: 

/. A straight line not through the center of inversion is transformed 
into a circle through the center of inversion. 

II. A straight line through the center of inversion is transformed 
into itself (and the line at infinity ). 

III. A circle not through the center of inversion is transformed into 
a circle not through the center of inversion (and the two minimum 
lines through the center of inversion). 

IV. A circle through the center of inversion is transformed into a 
straight line not through the center of inversion (and the two minimum 
lines through the center of inversion and the line at infinity*). 

V. A conic is transformed in general into a curve of fourth order 
through the circle points at infinity. 

VI. A conic through the center of inversion is transformed into 
a curve of third order through the circle points (and the line at 
infinity ). 

If we take the nonhomogeneous form (2) of the transformation 
and apply it to the equations 

ax + by + c = 0, 



we readily get theorems I-IV without the clauses in parentheses. 
It is in this simplified form that the theorems are often given, but 
they then fail to tell the whole story. 

Let us denote by / the transformation (1) and by M the trans 
formation III, 45. Then M~ l transforms the circle ^+y 2 =F 
into the unit circle, / carries out an inversion with respect to the 
unit circle, and M carries the unit circle back into the circle 
x z +y 2 =k 2 . The product of these three, namely MIM~ l , which is 



126 TWO-DIMENSIONAL GEOMETRY 

the transform of / by M, is an inversion with respect to the circle 
= 1? and is represented by the equations 

x = 1 



It is evident that a point P is transformed into a point P , where 
OP . OP f = k\ and that theorems I-VI still hold. 

If we desire an inversion with respect to a circle with center (#, >) 
and radius &, we may transform (3) by means of a transformation 
which carries into (a, b). The result is 

~j _ a *_(. x "" tt ) 

JL/ ^^ Ct 



Obviously theorems I-VI hold for (5). 

If the inversion (2) is written as an enlarged point-point trans 
formation of the form (3), 51, we have 



.., y 
y 



, = *y 

x 2 y z 

From this it is easy to compute that if p l and p z are the slopes of 
two curves through the same point, and if p( and p 2 are the slopes 
of the two transformed curves through the transformed point, then 



This shows that the angle between two curves is preserved by 
the transformation. A transformation which preserves angles is said 
to be conformed. Hence an inversion is a conformal transformation. 



CONTACT TRANSFORMATIONS IN THE PLANE 127 

EXERCISES 

1. Show that any circle through a point P and its inverse point P f 
is orthogonal to the circle of inversion. 

2. Show that a pencil of straight lines is transformed by inversion 
into a pencil of circles consisting of circles through two fixed points. 
Study the configuration formed by the inversion of a series of con 
centric circles and the straight lines through their common center. 

3. Show that parallel lines invert into circles which are tangent at 
the center of inversion. 

4. Show that the cross ratio of four points collinear with the center 
of inversion is equal to that of the transformed points. 

5. Show that a point P and its inverse point P f are harmonic con 
jugates with respect to the intersections of the line PP 1 and the circle 
of inversion. 

6. If a circle is inverted into a straight line, show that two points 
which are inverse with respect, to the circle go into two points which 
are symmetrical with respect to the line. 

7. Study the real properties of an inversion with respect to the 
imaginary circle x 2 -f- y* = 1. 

8. Show that an inversion is completely determined by two pairs 
of inverse points. 

9. From the theorem "four circles can be drawn tangent to three 
given lines " prove by inversion the theorem " four circles can be drawn 
tangent to three given circles which pass through a fixed point." 

10. From the theorem "two circles have four common tangent lines " 
prove by inversion the theorem " through a given point four circles can 
be drawn tangent to two given circles." 

54. Point-curve transformations. Consider now a transformation 
defined by the equation 

F(x v x^ x s , x{, x 2 , x 3 ) = 0, (1) 

where x f and x[ are point coordinates and F is a function homo 
geneous in both x i and #J, continuous in both sets of these variables, 
and possessing derivatives with respect to both. 

Let M be a point with the coordinates / { . If these coordinates 
are substituted for x f in (1) and held fixed, the resulting equation 
is that of a curve which we call an w -curve, the equation being 

f(.y v yy *i*i, O = o, (2) 

and we say that the point M is transformed into the m -curve. 



128 



TWO-DIMENSIONAL GEOMETRY 





FIG. 43 



We shall make the hypothesis that these w -curves form a two- 
parameter family of curves such that one curve of this family goes 
through any given point in any given direction. 

Let K be a point with the coordinates z{. This point will lie on the 
m -curve (2) if *(, y,^., ^, aj, <)= 0, (3) 

and all values of the ratios y l : y 2 : y z which can be determined 
from equation (3) will, if used in (2), determine an wi -curve 
through K . These values of y^ how 
ever, are given by any point M which 
lies on the curve 

F(x 1 ,x,,x a ,z(,4,4)=Q. (4) 
Call any curve defined by equation 
(4) a &-curve. We have, then, the 
following result: 

All points M which lie on a k-curve are transformed into m f -curves 
which pass through a point K (Fig. 43). 

We can say, then, that the k-curve is transformed into a point K!. 
In fact, the equation of a &-curve is found by holding x\ constant 
in (1), just as the equation of an w -curve is found by holding x i 
constant in the same equation. 

It is further evident that all k-curves which pass through a point M 
are transformed into points K which lie on the curve m . 

If any proof of this is necessary, it may be supplied by noticing 
that equation (3) is the condition that M should lie on k and 
that K should lie on m 1 . 

Consider now any curve c, 
not a &-curve, denned by the 
equations ^ = ^ (x> 

( 5 > 



FIG. 44 




The m -curves corresponding to 
points M on c form a one- 
parameter family of curves which in general have an envelope c , 
and the curve c is said to be transformed into the curve c ! . 

To follow this analytically let M^(x^ x# x s ) (Fig. 44) be the 
point on c corresponding to the value \ of X, and let M z be 



CONTACT TRANSFORMATIONS IN THE PLANE 129 

the point corresponding to the value X -f AX, the coordinates of 
M 2 being a^+A^, # 2 +A:r 2 , ^ 3 +A^ 3 . The two points M l and M 2 
are transformed into m( and m(, which intersect in a point K , the 
coordinates of which are given by the equations 

W C r T r ? * *r f \ 
\ v "yt *Si "v 21 z) v 



where the values of x i and Lx i are to be taken from (5). The 
point K corresponds to a &-curve through M^ and M 2 . 

Now let M 2 approach M^ The curve m( approaches the curve 
raj, and the point K approaches a limiting point T the coordinates 
of which are given by 

F(x^ x. 2 , x 8 , x{, x 2 , a)= 0, 

dF, , dF , dF , (7) 

dx+ dx + dx = Q, 

dx^ dx z dx 8 

where the values of x i and dx. are to be taken from (5). 

The point T[ is obviously the transformed point of , a Ar-curve 
tangent to c at M^ The locus of T is the curve c r , which cor 
responds to c. 

Equations (7) furnish a proof that c is tangent to m 1 at T . 
For, by differentiating the first of these equations and taking 
account of the second, we have 



which, as in 31, determines the direction of c . But this is just 
the equation which determines the direction of m(. The direction 
of c 1 is thus determined at the point T by the direction of m{. It 
is therefore determined by the point M l and the curve f, the latter 
being determined by the direction of c. Hence two curves c which 
are tangent are transformed into two curves c which are tangent. The 
transformation is therefore called a contact transformation. 

Suppose now that the transformation (1) is expressed in non- 
homogeneous Cartesian coordinates by the equation 



130 TWO-DIMENSIONAL GEOMETRY 

and let p be the slope of any curve c, and p 1 the slope 2Jj of 
dx ctx 

the transformed curve c . Then equations (6) and (8) are replaced 
in the present coordinates by 

dF dF 



which enable us to determine p and jt/ when z, y, x , and y ? are 
known. The last three equations, written together, 



/+*?-*. 

dx dy 



are called an enlarged point-curve contact transformation. If 
solved for z , y , and p 1 they may be written in the form 



(10) 



If, then, the point (a;, ?/) describes the curve re =/ i (X), ^ =/ 2 (\), 

f.YM 
we have jt? =*;J-^^^ an( i equations (10) give the transformed curve 



expressed in terms of the parameter X. 

An example of a point-curve transformation is found in the cor 
relations already discussed, since the equations (1), 42, may be 
written in the form 



Here the m -curves and the ^-curves are straight lines. If x i 
describes a curve c, the straight line m 1 envelops the transformed 
curve c r . If the correlation is expressed in Cartesian coordinates, 
it is readily put into the form (10). 



CONTACT TRANSFORMATIONS IN THE PLANE 131 

EXERCISES 

1. Express the general correlation in the form of equations (10). 

2. Place in the form of equations (10) the polarity by which a point 
is transformed into its polar line with respect to the circle x 2 -+- y* = 1. 

3. Find the curve into which the parabola y z = ax is transformed by 
the polarity of Ex. 2. 

4. Show that the curve into which the circle (x h) 2 -\-(y &) 2 = r 2 
is transformed by the polarity of Ex. 2 is a conic, and state the con 
ditions under which it is an ellipse, a parabola, or a hyperbola. Find 
the focus and directrix of the conic. 

5. Prove that by any polarity the order and the class of the trans 
formed curve is equal to the class and the order, respectively, of the 
original curve. 

6. Study the transformation 

=- 



-?, 

y 

and find the curve into which the circle # 2 +?/ 2 =l is transformed 
by it. 

7. Express in the form of equations (10) each of the types of 
correlations given in 42 and study them from this standpoint. 

55. The pedal transformation. As another example of a point- 
curve transformation we shall use homogeneous Cartesian coordi 
nates and take the equation 

<V 2 + / 2 > - x t x - y t y = 0. (1) 

If we take M as any point (x : y : ), the corresponding w -curve 
is in general a circle constructed on the line OM as a diameter. 
Exceptional points are the origin and the points at infinity. If M 
is the origin, the circle becomes the two minimum lines through 
the origin. If M is a point at infinity, not a circle point, the circle 
m splits up into the line at infinity and a straight line through 
perpendicular to OM. If M is a circle point /, the circle m f splits 
up into the line at infinity and the minimum line OL 



132 TWO-DIMENSIONAL GEOMETRY 

The &-curve corresponding to a point K r is in general a straight 
line through K and perpendicular to OK . Exceptions occur when 
K 1 is the origin or one of the circle points at infinity, in which 
cases the &-curve is indeterminate. If K is any point on the line 
at infinity but not a circle point, the &-curve is the line at infinity. 
If K is on a minimum line through 0, but not at infinity, the 
#-curve is the other minimum line through 0. A &-line does not 
in general pass through or the circle points at infinity. 

Conversely, any straight line which does not pass through the 
origin, and is neither the line at infinity nor a minimum line, is a 
&-line, the point K being the point in which the normal from 
meets the line. This may be seen by comparing the equation 
ax-\-by + ct = Q with (1), thus determining x : y : t^ aci bcia^+b 2 , 
which is the foot of the normal from to the line. 

Take any curve c. The tangent &-curve at any point M is 
the tangent line , and the point T r fe the foot of the perpen 
dicular from on T. Therefore ih&. transformed curve c of any 
curve c is the locus of the feet of the perpendiculars drawn from 
the origin to the tangent lines of c. The transformation is called 
the pedal transformation, and the point is the origin of the 
transformation. 

If the pedal transformation is expressed in Cartesian coordi 
nates as an enlarged point-curve transformation of the form (9), 
54, it becomes 



_ 
P 2y -y 

and these equations can be solved for x r , y , and p\ giving 



f 




CONTACT TKANSFOBMATIONS IN THE PLANE 133 

EXERCISES 

1. If Q is the pedal transformation with the origin 0, P a polarity 
with respect to any circle with the center 0, and R an inversion 
with respect to the same circle, prove the relations Q = RP, P = RQ, 
R = QP. 

2. Show that by a pedal transformation a parabola with its focus at 
the origin of the transformation is transformed into the tangent line 
at the vertex of the parabola. 

3. Show that by a pedal transformation an ellipse with its focus at 
the origin of the transformation is transformed into a circle with its 
diameter coinciding with the major diameter of the ellipse. State and 
prove the corresponding theorem for the hyperbola. 

Qu / \j 

4. Find the curve into which the ellipse -^ -f- ^ = 1 is transformed 

CL O 

by a pedal transformation with its origin at the center of the ellipse. 

56. The line element. With the use of Cartesian coordinates the 
contact transformations may be looked at from a new viewpoint 
by the aid of the concept of the line element. A line element may 
be denned as a point with an associated direction. More precisely 
let there be given three numbers (#, y, p), where the numbers 
x and y are to be interpreted as the usual Cartesian coordinates 
of a point in the plane and p is to be interpreted as the slope 
or direction of a line through the point. Then the three quanti 
ties taken together define a line element. A line element may 
be roughly represented by plotting a point M and drawing a short 
line through M in the direction p, but this line must be con 
sidered as having no length just as the dot which represents M 
must be considered as without magnitude. There are oo 3 line 
elements in the plane out of which we may form a one-dimensional 
extent of line elements by taking #, y, and p as functions of a 
single parameter; thus, 



P = 

There are two types of one-dimensional extents : 

TYPE I. The functions / x (X) and/ 2 (X) may reduce to constants. 
In this case the one-dimensional extent consists of a fixed point 
with all possible directions associated with it. 



134 TWO-DIMENSIONAL GEOMETRY 

TYPE II. The point (a?, y) may describe a curve the equations 
of which are the first two of (1). Then the third equation of (1) 
associates with every point of that curve a certain direction. 

It is obviously convenient that the direction associated with each 
point of the curve should be that of the tangent to the curve. The 
necessary and sufficient condition for this is that by virtue of (1) 
we should have dx pdy = 0. 

A one-dimensional extent of line elements defined by equation (1) 
shall be called a union of line elements when it satisfies the con 
dition dx pdy = 0. It is evident that the first type of extents 
always satisfies this condition and that the second type satisfies the 
condition when the direction of each element is that of the curve 
on which the point of the element lies. 

Two unions of line elements have contact with each other if they 
have a line element in common. Two unions of the first type have 
contact, therefore, when they coincide ; one of the first type has con 
tact with one of the second when the point of the first lies on the 
curve of the second; and two elements of the second type have 
contact when their curves are tangent in the ordinary sense. 

Any transformation of line elements defined by the equations 



tf=f*(MP\ (2) 

P =/B( X I y*p)> 

where the functions are bound by the condition 



(3) 

where p is not identically zero, is called a contact transformation. 

It is clear that by such a transformation a union of line ele 
ments is transformed into a union of line elements and that two 
unions which are in contact are transformed into two which are 
in contact. 

The enlarged point-point transformation (3), 51, and the 
enlarged point-curve transformation (9), 54, are cases of the 
general contact transformation (2). In fact, any contact trans 
formation may be reduced to one of these cases. To show this 
let us proceed to deduce from (2) equations which are free from 
p and p . Two cases only can occur. 



CONTACT TRANSFORMATIONS IN THE PLANE 135 

CASE I. The first two equations in (2) may each be free from JP. 
Then equation (3) gives the condition 



which must be true for all values of the ratios dx : dy. Hence we have 

tt -j^a^s 

dy dy 



whence, by eliminating p and solving for p , we have the result 
that the contact transformation (2) is in this case of the form 



= 



which is exactly that of (3), 51. 

By this transformation any one-dimensional extent of line ele 
ments which form a union of the first type is transformed into a 
union of the first type, and any union of the second type is trans 
formed into a union of the second type. 

CASE II. At least one of the first two equations in (2) contains^. 
It is then possible to find one, but only one, equation free from 
p and p . Let that equation be 

^(^#a/,y)=0. 

From this equation we find 

dF, , dF, , dF, , , dF, , A 
dx + dy + : dx f + dy 1 =0, 

dx dy dx 1 dtf 

which must be identical with (3). By comparison we find 

dF __ dF _ d_F_ _dF 
~dx~ ~dy " ^dx 1 ~ dy 

PP -P -P 1 



136 TWO-DIMENSIONAL GEOMETRY 

from which p and p can be found, with the result that the contact 
transformation (2) can in this case be put into the form 



which is exactly that of (9), 54. 

By this transformation any union of the first type is transformed 
into a union of the second type, each element of the former being 
transformed into an element of the latter. 

As an example consider the transformation 




If written in the form (5) this becomes 



The geometrical meaning of these equations is simple. Any line 
element (x, y, p) is transformed into a line element (V, y\ p ~) so 
placed that the point (V, y ) is at a distance k from the point (V, # ), 
and the line joining (V, y 1 ) to (#, y) is perpendicular to the line 
element. A transformed line element is parallel to the original 
element. Otherwise stated, each line element is moved parallel 
to itself through a distance k in a direction perpendicular to the 
direction of the element. Each line element is therefore trans 
formed into two line elements. A union of the first type, consist 
ing of line elements through the same point, is transformed into a 
union consisting of the line elements of a circle with that point as 
a center and a radius k. Any curve c is transformed into two 
curves parallel to c at a normal distance k from c. 

This transformation is sometimes called a dilation, suggesting 
that each point of the plane is dilated into a circle. 



CONTACT TRANSFORMATIONS IN THE PLANE 137 

EXERCISES 

1. Show that the transformation 



y =xp-y, 
p =x, 

is a contact transformation and study its properties. 
2. Show that the transformation 



p =p, 
is a contact transformation and study its properties. 

3. Show that any differential equation of the form f I x, y, -^- ) = 

may be written in the form f(x, y, p) = and considered as defining a 
doubly infinite extent of line elements. To solve the equation is to 
arrange the elements into unions of line elements. In general, the solu 
tion consists of a family of curves. Any union formed by taking one 
element from each curve of a family is a singular solution. Note that 
an equation f(x, y) = can also be interpreted in this way, and that 
the family of solutions consists of points on the curve f(x, y) = with 
all the line elements through each, while the singular solution is the 
curve /(x, y) = with its tangent elements. 

4. Study the differential equation y px = in the light of Ex. 3. 
Show that the singular solution is the one-dimensional extent of line 
elements which consists of all elements through the origin. 

5. Apply to Ex. 4 the dilation x =x -- , y = y-\ 

I 



p =p . Show that the differential equation becomes y p x Vl+^? 2 = 0. 
What becomes of the singular solution and the family of solutions ? 

6. Study Clairaut s equation, y=px+f(p), by the method of 
Ex. 3 and show geometrically that the family of solutions consists of 
the straight lines y = ex +/(c). What is the singular solution ? Apply 
to the variables in the equation the transformation xx + yy = 1 and 
determine the effect on the equation and its solutions. 



CHAPTER IX 



TETRACYCLICAL COORDINATES 

57. Special tetracyclical coordinates. We shall discuss in this 
chapter a system of coordinates especially useful for the treat 
ment of the circle. These coordinates are not dependent upon the 
Cartesian coordinates, though they are often so presented. On the 
contrary they may be set up independently by elementary geometry 
for real points and then extended to imaginary and infinite points 
in the usual manner. It is therefore not to be expected that the 
geometry in the imaginary domain and at 
infinity should agree in all respects with 
that obtained by the use of Cartesian 
coordinates. 

The coordinates we are to discuss are 
called tetracyclical coordinates, and we 
begin, for convenience, with a special type. 

Let OX and OY (Fig. 45) be two 
straight lines of reference intersecting at 
right angles at 0, and let P be any real point of the plane. Let 
MP and NP be the distances of P from OX and Y, respectively, 
taken with the usual convention as to signs, and let OP be the 
distance of P from 0, taken always positive. Then the special 
tetracyclical coordinates of P are the ratios 



N 



M 
FIG. 45 



= OP :NP: 



(1) 



from which it follows that the quantities x are connected by 
the fundamental relation 

(x)=xt+xt-x0 t =Q. (2) 

It is obvious that to any real point corresponds one set of coor 
dinates and, conversely, to any real set of the ratios x^x^ix^.x^ 
which satisfy the relation (2), and for which x 4 3 s 0, corresponds 
one real point P. We extend the coordinate system in the usual 

138 



TETRACYCLICAL COORDINATES 



139 



manner by the convention that any set of ratios satisfying (2) 
shall define a real or an imaginary point of the plane, the ratios 
0:0:0:0 being of course unallowed. 

As the real point P recedes from 0, the ratios approach a limit 
ing set of values 1:0:0:0. To see this we write equation (1) in 
the form 






NP MP 



= 1: 



OP OP OP 

cos 6 sin 6 



OP OP 



where 6 = the angle MOP. The limit of the ratios of x i is there 
fore 1:0:0:0. Hence we say that by the use of the special tetra- 
cyclical coordinates the plane is regarded as having a single real point 
at infinity. This point, however, is not the only one which must 
be considered at infinity, as will appear later. 

58. Distance between two points. Let C (y^ -y^y ^y^) and 
P (x^: x 2 :x 3 : # 4 ) (Fig. 46) be two real points, and let d = CP, the 
distance between them. Then, by trigonometry, 

d*= OP 2 + ~OC 2 - 2 OP OC cos (^- 2 ), 

where 1= =the angle XOP and 2 = the angle XOC. But from 
the definition of the coordinates and y 

from the relations 



OPcos0=- 2 , 
x, 



OP sin =- 8 , 
X A 



OC cos = 2 , OC sin = 

* 



the above equation can be written 




FIG. 46 



^ 



This equation, obtained by the use of real points, is now taken 
as the definition of the distance between imaginary points. 
Equation (1) can be written 



2 



140 TWO-DIMENSIONAL GEOMETRY 

where in accordance with the usual notation co(x, y) denotes the 
polar* of the form o>(V). 

From (3) it appears that d oo when y^ or when # 4 = 0. Hence 
the locus of the points at infinity is defined by the equation x 4 = 0. 

Since always CD (V) = 0, the points at infinity satisfy also the con 
dition x% + x = 0, from which it appears that the point 1:0:0:0 
is the only real point at infinity, as we have already seen. The 
nature of the locus at infinity will appear later. 

59. The circle. If we take the usual definition of a circle, the 
equation of a circle with center y { and radius r can be written from 

y? 1 -*y? t -2y? t +<y l -f*yjx t =0. (1) 

This is of the type 



and the relations between the coefficients a. and the center and 
radius of the circle are readily found. For we have by direct 
comparison of (1) and (2) 

P a i=y# P a i=-2y* P a B=- 2 ^ P a *=yi- r \ 

From these and the fundamental relation y%-\-y% y^J^ we 
easily compute the following values : 



(3) 



o_ 



* A homogeneous polynomial is called a form. The general quadratic form in 
n variables is 



n , 

and the bilinear form *aaM!/t 






is called the polar form of (1). If by a linear transformation of the variables xl 
the form (1) is transformed into 



its polar is transformed into 



TETRACYCLICAL COORDINATES 141 

which give the coordinates of the center and the radius of the 
circle in terms of the coefficients a i of equation (2). 

These results, obtained primarily for real circles, are now gen 
eralized by definition as follows : 

Every linear equation of the form (2) represents a circle, the center 
and the radius of which are given by equations (3). 

We may classify circles by means of the expression for the radius. 
For that purpose let us denote the numerator of r 2 in (3) by rj (V); 

that is > 17 (V) = <+ al- 4 a,a,. (4) 

We make, then, the following cases : 

CASE I. ij (a) = 0. Nonspecial circles. 

Subcase 1. a l = 0. Proper circles. Equation (2) is reducible to 
(1) and represents the locus of a point at a constant distance from 
a fixed point. Neither center nor radius is necessarily real, but the 
center is not at infinity and the radius is finite. The circle does not 
contain the real point at infinity, since 1:0:0:0 will not satisfy 
equation (2). 

Subcase 2. a^= 0. Ordinary straight lines. The radius becomes 
infinite and the center is the real point at infinity. The equation 
may be written, by 57, in the form 

a z NP + a & MP + 4 = 0, + al * 0) 

which, as in Cartesian geometry, is a straight line. This line 
passes through the real point at infinity. In fact, the necessary 
and sufficient condition that equation (2) should be satisfied by 
the coordinates of the real point at infinity is that a l = 0. Hence 
an ordinary straight line may be defined as a nonspecial circle which 
passes through the real point at infinity. 

CASE II. 77 (a) = 0. Special circles. 

Since a 2 2 + # 3 2 = 4 a^a, the coordinates of the center may be written 



Subcase 1. a^ 0. Point circles. The radius is zero and the coordi 
nates of the center are those of a point not at infinity. The center 
may be any finite point. It is obvious that if the center is real, it is 
the only real point on the circle, and hence the name " point circle." 
The point circles do not pass through the real point at infinity. 



142 TWO-DIMENSIONAL GEOMETEY 

By (2), 58, the equation of a point circle may be written 

w(x, #)=0, 

where (?/) 0. Comparing with (4), we see how the equation 
77 (a) = may be deduced from CD (y) = 0. 

Subcase 2. a 1 = 0. Special straight lines. The radius becomes inde 
terminate, and the center, given by (4), becomes 2 a^ : a z : a s ^ 0, 
which is a point at infinity. The special straight lines pass through 
the real point at infinity.- In fact, a special straight line may be 
defined as a special circle which passes through the real point at infinity. 

We have seen that the locus of all points at infinity is # 4 = 0, 
which is the equation of a circle belonging to the case now being 
considered, and with its center at 1 : : : 0. Hence we say : 

The locus at infinity is a special straight line whose center is the 
real point at infinity. 

EXERCISES 

1. Consider the point circle 3^=0. Show that it is made up of 
two one-dimensional extents ("threads") expressed by the equations 
a?! : x 2 : x s : x : 1 : i : \, where X is an arbitrary parameter. Show 
that these threads have the one point 0:0:0:1 in common, but that 
neither can be expressed by a single equation in tetracyclical coordi 
nates. Hence note the difference between this locus and that expressed 
by ar 2 + y 2 = in Cartesian coordinates. 

2. As in Ex. 1, show that the special circle x 4 = is composed of two 
threads having the real point at infinity in common. 

3. Examine the special circles x 2 -\- ix 3 = and x 2 ix s = and show 
that these two and the two in Exs. 1 and 2 are made up of different 
combinations of the same four threads. 

4. Show that any special circle is made up as is the circle in Ex. 1. 

60. Relation between tetracyclical and Cartesian coordinates. If we 

introduce Cartesian coordinates, by which, in Fig. 45, 



there exists for any real point of the plane the following relation 
between the special tetracyclical coordinates and the Cartesian 
coordinates : ox =x 2 +if 



TETRACYCLICAL COORDINATES 143 

These equations, derived for real points of the plane at a finite 
distance from 0, can now be used to define the relation between 
the imaginary and infinite points introduced into each system of 
coordinates. 

There appear, then, exceptional points. In the first place, we 
notice that the tetracyclical coordinates take the unallowed values 
0:0:0:0 when x 2 + y* = 0, t = 0. That is, the circle points at 
infinity necessary in the Cartesian geometry have no place in the 
tetracyclical geometry. Furthermore, any point on the line at 
infinity t = 0, other than a circle point, corresponds to the real 
point at infinity 1:0:0:0 in the tetracyclical coordinates. 

If the tetracyclical coordinates are given, the Cartesian coordi 
nates are obtained through the equations xt : yt : t 2 = x 2 : x 3 : x^. These 
equations will determine a single point on the Cartesian plane 
unless x 2 = x s = # 4 = 0. In this case t = and the ratio x : y is 
indeterminate. That is, the real point at infinity in tetracyclical 
coordinates corresponds to the entire line at infinity in Cartesian 
coordinates. Any other point on the tetracyclical locus at infinity 
# 4 = has coordinates of the form x l : 1 : i : 0, and no Cartesian 
coordinates can be found corresponding to these values. 

Hence, in Cartesian coordinates we find certain points, the circle 
points at infinity, which do not exist in tetracyclical coordinates, and 
in tetracyclical coordinates we find certain points, the imaginary points 
at infinity, which do not exist in the Cartesian coordinates. We also 
find that the real point at infinity in tetracyclical coordinates corre 
sponds to the entire line at infinity in Cartesian coordinates, and, con 
versely, that any point at infinity in Cartesian coordinates corresponds 
to the real point at infinity in tetracyclical coordinates. With these 
exceptions the relation between the coordinates is one to one. 

The exceptional cases bear out the statements in 3 and 4 as 
to the artificial nature of the conventions as to imaginary points 
and points at infinity. Since the Cartesian coordinates are more 
common, there is some danger of thinking that the conventions 
there made are in some way essential. The discussion of this text 
shows, however, that the tetracyclical conventions may be made 
independently of the Cartesian ones, and the geometry thus deduced 
is equally as valid as the Cartesian. As long as either set of 
coordinates is used by itself, the difference in the conventions is 



144 TWO-DIMENSIONAL GEOMETRY 

unnoticeable. It is only when we wish to pass from one set of 
coordinates to the other that we need to consider this difference. 
61. Orthogonal circles. Consider two proper circles with real 
centers C a and C b and real radii r a and r b , intersecting in a real 
point P. Then, if (r a , r b ) is the angle between the radii C a P and 
(7 6 P, and d is the length of the line C a C w we have, from trigonometry, 



cos(r a , r b ) = - 2 - 



But the angle between the circles is either equal or supple 
mentary to the angle between their radii. Hence, if we call 6 the 
angle between the circles we have 

cos 6 = - 



If the equations of the two circles are 

/y.r_l_/y r_l_/Y r_l_/Y r f~\\ 

and b& + b& + b s x s + b^ = (2) 

respectively, the formula for the angle may be reduced by (3), 59, 
and (4), 59, to the form 



f .. f 



aJ+ a*- 4 A 6 2 2 + J s 2 - 4 
or, more compactly, 




cos<? = *=, (3) 



where ?y(, 5) is the polar of 77 (a). 

This formula, which has been obtained for two real proper circles 
intersecting in a real point, is now taken as the definition of the 
angle between any two circles of any types whose equations are 
given by (1) and (2). We leave it for the reader to show that if 
one or both of the circles is a real straight line, the definition 
agrees with the usual definition. 

The condition that two circles should be orthogonal is then 

l(,*) = 0. (4) 

If the circle (1) is a special circle, the coordinates of its center 
have been shown to be 2 a^ : a 2 : a & : 2 a^ and equation (4) is the 



TETRACYCLICAL COOBDINATES 145 

condition that this center should lie on (2). Hence a special circle, 
whether a point circle or a special straight line, is orthogonal to 
another circle when and only when the center of the special circle lies 
on the other circle. 

We might equally well say that a special circle makes any angle 
with a circle on which its center lies, since in such a case cos in 
(3) is indeterminate. 

It is possible in an infinity of ways to find four circles which 
are mutually orthogonal. For if 

2<W= (5) 

is any circle, the circle 

2**= (6) 

may be found in oo 2 ways orthogonal to (5), since the ratios b. have 
to satisfy only one linear equation of the form (4). Circles (5) 
and (6) being fixed, the circle 

X<V = (7) 

may be found in an infinite number of ways orthogonal to (5) and 
(6), since the ratios c { have to satisfy only two linear equations. 
Finally, the circle 

*- 

may be found orthogonal to (5), (6), and (7) by solving three 
linear equations for e { . 

It is geometrically evident that at least one of these circles is 
imaginary. 

EXERCISES 

1. Prove, as stated in the text, that formula (3) gives the ordi 
nary angle in the cases in which one or both of the circles is a 
straight line. 

2. Prove that a special circle is orthogonal to itself. 

3. What is the angle between a special circle and another circle not 
through its center ? 

4. Prove that the circles x l x 4 = 0, x 2 = 0, x s = are mutually 
orthogonal and find a fourth circle orthogonal to them. 

5. Prove that a^ = 0, # 2 = 0, cc 3 = are mutually orthogonal. Can a 
fourth circle be found orthogonal to them ? Explain. 



146 TWO-DIMENSIONAL GEOMETRY 

6. Find all circles orthogonal to the circle at infinity # 4 = 0. 

7. Find the equations of all circles orthogonal to the point circle 
x l = 0. How do they lie in the plane ? 

8. Find the equations of all circles orthogonal to the real proper 
circle x l x 4 = 0. 

9. Show that all circles whose coefficients a { satisfy a linear equation 



are in general orthogonal to a fixed circle and find that circle. 
62. Pencils of circles. Consider two circles 

<*&+ a fr + V + a t x t = CO 

5^+5^+^+6^=0. (2) 

With reference to them we shall prove first the following 
theorem : 

/. Any two circles intersect in two and only two points. These points 
may be coincident, in which case the circles are said to be tangent. 

To prove this we note that if equations (1) and (2) are inde 
pendent, at least one of the determinants, aJbj ajb^ must be different 
from zero. Hence we can solve for one pair of variables, x i and Xj, 
, in terms of the other two. For example, we may find from (1) and 
(2) # 1 =<? 1 # 3 + Vv X 2 = C 3 X 3~^~ C 4 X 4 ^ these values are substituted 
in the fundamental relation o>(V)=0, there results a quadratic 
equation in x 3 and x^. This determines two values of x s : x^ and 
from each of these the ratios x l : x 2 are determined. This proves 
the theorem. 

It is evident that the circle points at infinity which are intro 
duced as a convenient fiction in Cartesian geometry do not appear 
here. In Cartesian geometry it is found that there are always two 
sets of coordinates which satisfy the equation of any circle, and we 
are consequently led to declare that all circles pass through the 
same two imaginary points at infinity. By the use of tetracyclical 
coordinates there are no two points at infinity common to all 
circles. In fact the circle (1) meets the locus at infinity x^= in 
the two points a z T a t i : a^ : ia l : 0, which are not the same for 
all circles. 



TETKACYCLICAL COORDINATES 147 

Theorem I holds of course for the case in which the circles are 
straight lines, one of the points of intersection being always the real 
point at infinity. Two straight lines which are tangent at the real 
point at infinity are parallel lines in the Cartesian geometry. 

Consider now the equation 



where X is an arbitrary parameter. For any value of X (3) defines 
a circle which passes through the points common to (1) and (2) 
and intersects (1) and (2) in no other point. The totality of the 
circles corresponding to all values of X forms a pencil of circles. 

If (1) and (2) are real circles, the pencil (3) may be of one of 
the following types : 

(1) proper circles intersecting in the same two real points ; 

(2) proper circles intersecting in the same two imaginary points ; 

(3) proper circles tangent in the same point ; 

(4) proper concentric circles ; 

(5) a pencil of intersecting straight lines ; 

(6) a pencil of parallel straight lines. 

II. In any pencil of circles there is one and only one straight line, 
unless the pencil consists entirely of straight lines. 

The condition that (3) should represent a straight line is 

i+ X6 i= 0, 

which determines one and only one value of X unless both a l and 
b l are zero. In the latter case all circles defined by (3) are straight 
lines. This proves the theorem. 

The straight line of the pencil is called the radical axis of any 
two circles of the pencil. Its equation is 

(A- AX+0*A- AX+(A- AX= - 

This is a special line when 



If the circles (1) and (2) are real and proper, the last equation 
can be satisfied only when 



148 TWO-DIMENSIONAL GEOMETRY 

and the equations (1), (2), and (3) represent concentric circles, 
and the radical axis is the line at infinity x^ 0. 

In all other cases the radical axis of two real circles is a real 
straight line. 

///. In any pencil of circles there are two and only two (distinct or 
imaginary) special circles, unless the pencil consists entirely of special 
circles. 

By 59 the condition that (3) should be a special circle is 



or 97 (a) + 2 \rj (a, b) + \ 2 rj (5) = 0. 

This equation determines two distinct or equal values of X 
unless it is identically satisfied. Hence the theorem is proved. 

If the pencil is defined by two real proper circles, the special 
circles are point circles, since by II there is only one straight line 
in the pencil and that is real and nonspecial. It is not difficult to 
show that if the circles of the pencil intersect in real points, the 
special circles have imaginary centers ; if the circles of the pencil 
intersect in imaginary points, the special circles have real centers ; 
and if the circles of the pencil are tangent, the centers of the special 
circles coincide at the point of tangency. 

IV. A circle orthogonal to two circles of a pencil is orthogonal to all 
circles of the pencil. 

Let ^\o.x.= be orthogonal to (1) and (2). Then 

T?(C, fl)=0, 17(0, &) = 0; 
whence 77 (c, a -f- X6) = 97 (<?, a) + XT; (<?, 5) = 

for all values of X. This proves the theorem. 

It follows from this and 61 that a circle orthogonal to all 
circles of a pencil passes through the centers of the special circles 
of the pencil, and, conversely, a circle through the centers of 
the special circles is orthogonal to all circles of the pencil. If the 
pencil has only one special circle, the orthogonal circles can be 
determined as circles which pass through the center of the special 
circle and are ortliogonal to one other circle of the pencil, say the 
radical axis. 



TETRACYCLICAL COORDINATES 



149 



These considerations lead to the following theorem : 
V. For any pencil of circles there exists another pencil such that all 
circles of either pencil are orthogonal to all circles of the other, and 
any circle which is orthogonal to all circles of one pencil belongs to the 
other. The points common to the circles of one pencil are the centers 
of the special circles of the other. 

Fig. 47 shows such mutually orthogonal pencils. 




FIG. 47 

EXERCISES 

1. Show that two real circles intersect in two real distinct points, 
are tangent, or intersect in two conjugate imaginary points according as 



2. Show that the point circles in a pencil of real circles have real and 
distinct, conjugate imaginary, or coincident centers, according as the 
circles of the pencil intersect in conjugate imaginary, real and distinct, 
or coincident points. In the last case show that the centers of the point 
circles coincide with the point of tangency of the circles of the pencil. 

3. Show that circles which intersect in the same two points at infinity 
are concentric. 

4. Prove that the radical axis of a pencil of circles passes through 
the centers of the circles of the orthogonal pencil. 

5. Prove that the radical axes of three circles not belonging to the 
same pencil meet in a point. 

6. Take ^a^= 0, ^T&.x t -= 0, ^Pc^.= 0, any three circles not be 
longing to the same pencil, and show that ^V (a,. -}- \b { + /*c { ) x f = 
defines a two-dimensional extent of circles (a circle complex) consisting 
of circles orthogonal to a fixed circle. Discuss the number and position 
of the point circles, the straight lines, and the special lines of a complex. 

7. Show that the totality of straight lines form a complex. To what 
circle are they orthogonal ? 

8. Show that circles common to two complexes form a pencil. 



150 TWO-DIMENSIONAL GEOMETRY 

63. The general tetracyclical coordinates. Let us take as circles 
of reference any four circles not intersecting in the same point 
and the equations of which, in the special tetracyclical coordinates 
thus far used, are 



and let us place 

n ^T =2 n T, -4- H v, -\~ n T. -\- n T. 

(i) 



Since the four circles do not meet in a point their equations 
cannot be satisfied by the same values of #,., and therefore the 
determinant of the coefficients in (1) does not vanish. Therefore 
the equations can be solved for x t with the result 






where A f is the cofactor of a i in the determinant of the coefficients 
of (1), B f the cofactor of $, etc. 

The relation between the ratios x-.x-.x-. x. and X : JT : X : X 

1234 12o4 

is therefore one to one, and the latter ratios may be taken as the 
coordinates of any point. These are the most general tetracyclical 
coordinates. 

A geometric meaning may be given to these coordinates as 
follows : 

If the circle with the Cartesian equation 

a(x*+y^ + lx + cy + d = Q 

is a real proper circle, and the point P (x, y) is a real point outside 
of it, then the expression 



is proportional to the power of P with respect to the circle ; that is, 
to the length of the square of the tangent from P to the circle. If 



TETRACYCLICAL COORDINATES 151 

P is a real point inside the circle, the power may be defined as the 
product of the lengths of the segments of any chord through P. 

Also, if 

bx + cy + d = 

is a real straight line, the expression 

bx + cy + d 

is proportional to the length of the perpendicular from any real 
point to the line. 

By virtue of 60 these relations hold for a linear equation in 
tetracyclical coordinates. Of course if the points, circles, or lines 
involved are imaginary, the phraseology is largely a matter of 
definition. We may say, then : 

The most general tetracyclical coordinates of a point consist of the 
ratios of four quantities each of which is equal to a constant times the 
power of the point with reference to a circle of reference, or, in case 
the circle of reference is a straight line, to a constant times the length 
of the perpendicular from the point to the line.* 

By means of (1) the fundamental relation co (x) = goes over 
into the new fundamental relation 



o> (3) 

and the polar equation o> (x, y) = becomes 

0(.z;r)=2X*;r=o, (4) 

where the determinant | % | does not vanish. 

The real point at infinity has now the coordinates Xj Xj X 3 : X 4 
= aj Pj 7 1 : Sj, and hence by a proper choice of the circles of 
reference may be given any desired coordinates. The locus at 
infinity has the equation 



* Some authors prefer to define the coordinate as the quotient of the power of the 
point divided by the radius, since this quotient goes over into twice the length of 
the perpendicular from the point to a straight line when the radius of the circle 
becomes infinite. This definition fails if the circle of reference is a point circle 
when the corresponding coordinate is the square of the distance of the point from 
the center of the circle. Since the constant which may multiply each coordinate is 
arbitrary, we prefer the definition in the text. 



152 TWO-DIMENSIONAL GEOMETRY 

A circle with the equation 

^+v > +v*+v- 

has in the new coordinates the equation 

A^ + A^ + A^ + AtX^Q, 
where p^= o^ + ^^-h 7^3+ M*> 



By virtue of these relations the condition for a special circle 
i) (a) = becomes a new relation 

H(J) = 5)^^=0, (6) 

and the condition rj (a, 5) = for orthogonal circles becomes 

0. (7) 



The form H (A) may be computed directly from II (X) as follows : 
By formulas (4) and (2), 58, the equation of a point circle 
with the center Y. is A 

JT 1*5 v 



Hence, if A 

is a point circle, we must have 

pA, = a {l Y, + a i2 Y 2 + a F, + a ft F 4 . (8) 

These equations can be solved for Y i since the determinant a ik \ 
does not vanish. But Y t being the coordinates of a point must sat 
isfy the fundamental relation (3). Substituting, we obtain a rela 
tion between the A s to be satisfied by any point circle. This can 
be nothing else than the condition 



By virtue of (8) we have, accordingly, 



But (8) can be written o-A t = 

Hence we have H = Ktt ( F). (9) 



TETKACYCLICAL COORDINATES 153 



Also the form fl (X) may be computed from the form H 04) as 
follows : If A is a point circle, equation (7) expresses the condition 
that the center of A should lie on a circle B. But if X. are the 
coordinates of the center of A, this condition is 



Hence, by comparison with (7), 

^=M,+M 2 +M3+M4- (10) 

Since A is a point circle its coefficients A i satisfy (6). Therefore, 
if equations (10) are solved for A. and the result substituted in 
(6), we have a relation satisfied by the coordinates of any point. 
This can only be 



By virtue of (10) we have, accordingly, 



But (10) can be written trX.= - 



Hence we have O = Tffl (A). (11) 

64. Orthogonal coordinates. Particular interest attaches to the case 
in which the four circles of reference are mutually orthogonal. If 
the circle X.= is orthogonal to the circle X k = 0, we have, from (7), 
63, b ik = 0. Therefore, for an orthogonal system of coordinates 
WP havp 

H oi)= Mi 2 + Ml + Ml + M* 

Equations (10), 63, give 

pX i =k i A { , 
whence the fundamental relation for the point coordinates is 



Without changing the coordinate circles it is obviously possible 
to change the coefficients in (1), 63, so that & f =l. Then we have 



154 TWO-DIMENSIONAL GEOMETRY 

A special case is obtained by placing 



where x. are the special coordinates of 57. The four circles of 
reference are a real circle with center at and radius 1, two per 
pendicular straight lines through 0, and an imaginary circle with 
center at and radius i. 

65. The linear transformation. Let x { be any set (special or 
general) of tetracyclical coordinates where a>(V)=0 is the fun 
damental relation, and consider the transformation denned by the 

equations 

px{ = a^ + a l2 x 2 + a u x s + a^, 

ax ax o^, 



where the determinant of the coefficients \a ik does not vanish and 
where x\ satisfies the same fundamental relation as x? 

By means of (1) any point x i is transformed into a point a/., and 
since the equations can be solved for x t , the relation between a 
point and its transformed point is one to one. 

By means of (1), also, any circle 

Vi+V 2 +V 3 +%= 
is transformed into the circle 

a(x[ + a 2 x 2 + X + X = > 
where pa[ = A^a^ -f A i2 a z + A i8 a s + A^a^, 

Now, if y i is a fixed point, # f a variable point, and y\ and x( the 
transformed points respectively, the equation 

"><> y)= 
is transformed into the equation 

o><y, y)=o, 

since the equation o> (V) = is transformed into co (V) = 0. 



TETRACYCLICAL COORDINATES 155 

That is, by the transformation (1) special circles are transformed 
into special circles, the center of each special circle being transformed 
into the center of the transformed circle. 

It follows from the above that nonspecial circles are transformed 
into nonspecial circles, for if a nonspecial circle were transformed 
into a special circle, the inverse transformation would transform a 
special circle into a nonspecial circle, and since the inverse trans 
formation is also of the form (1), this is impossible. 

We may accordingly infer that by the transformation (1) the 
equation rj (a) = is transformed into itself. 

We may distinguish between two main classes of transformations 
of the form (1) according as the real point at infinity is invariant 
or not. The truth of the following theorem is evident : 

If a linear transformation leaves the real point at infinity invariant, 
every straight line is transformed into a straight line and every proper 
circle into a proper circle. If a linear transformation transforms the 
real point at infinity into a point and transforms a point into 
the real point at infinity, any straight line is transformed into a circle 
through 0, and any circle through is transformed into a straight line. 

Since, as we have seen, the equation 77 (a) = is transformed into 
itself, we may write r;(V) = kr](cf), the value of k depending on 
the factor p in (1). With the same factor we have 77 (> ) = krj (6) 
and ?; (V, 5 ) = Jcr/ (a, ). Hence by (3), 61, the angle between 
two circles is equal to the angle between the two transformed 
circles. The linear transformation is therefore conformal. 

66. The metrical transformation. We shall prove first that any 
transformation of the metrical group can be expressed as a linear 
transformation of tetracyclical coordinates. 

We have seen in 45 that a transformation .of the metrical group 
is a linear transformation of the Cartesian coordinates x and y 
together with the condition (V 2 + y 2 ) = # 2 (# 8 + ?/ 2 ). It follows from 
this that the transformation can be expressed as a linear transfor 
mation of the special coordinates of 57. But the general tetra 
cyclical coordinates are linear combinations of the special ones. 
Hence the theorem is proved. 

Since a metrical transformation transforms straight lines into 
straight lines, it must leave the real point at infinity invariant. 



156 TWO-DIMENSIONAL GEOMETRY 

Conversely, any linear transformation of tetracyclical coordinates 
which leaves the real point at infinity invariant is a transformation of 
the metrical group. This may be shown as follows : 

If the real point at infinity is invariant, the locus at infinity is 
transformed into itself, since it is a special circle with its center at 
the real point at infinity. Therefore any linear transformation of 
general tetracyclical coordinates which leaves the real point at infinity 
invariant is equivalent to a transformation of the special coordinates 
of 57, which leaves the point 1:0:0:0 invariant and transforms 
the locus # 4 = into itself ; that is, to a transformation of the form 

px( = a^+ a l2 x 2 + is z,+ M * 4 , 



Since x? + af - x[x[ = k 2 (x\ + xl - a^), (2) 

we have, for the coefficients, the conditions 



+ i = < + i = n = -I 



(3) 



Now the last three equations of (1) are equivalent to the equa 
tions in Cartesian coordinates 



and the conditions imposed on the coefficients are exactly those 
necessary to make this a metrical transformation. The first equa 
tion in (1) is a consequence of the last three equations in (1) and 
the condition (2). In fact, the coefficients # 22 , # 23 , o: go , and # 33 may 
first be determined to satisfy equations (3), the coefficients # 24 and 
a^ may be assumed arbitrarily, and the coefficients # n , a l2 , # 13 , 
and a u are then determined by (3). This proves the theorem. 

67. Inversion. Two points P and P f are inverse with respect to a 
nonspecial circle C if every circle through P and P r is orthogonal 
to C. From this it follows that if C is a straight line two inverse 



TETRACYCLICAL COORDINATES 157 

points are symmetrical with respect to that line ; that is, the straight 
line PP is perpendicular to C and bisected by it. By a limit process 
it is natural to define the inverse of a point on the straight line C 
as the point itself. 

If C is a proper circle with radius r and center A (Fig. 48), the 
inverse of A is the real point at infinity, since the circles which 
pass through A and the real point at infinity are straight lines 
perpendicular to C. If P is not at A 
nor on (7, the straight line PP must 
pass through A, since that line is a 
circle through P and P which by defi 
nition must be orthogonal to C. Take 
now the point M midway between P 
and P so that 




and with M as a center construct a 

circle through P and P . If R is the radius of this circle, 



By squaring the last two equations and subtracting one from the 
other, we have AM*-X* = AP -AP . 



But the condition for orthogonal circles gives 
R*+r 2 -AM 2 = 0. 

Hence we have as the condition satisfied by two inverse points 
with respect to a circle with radius r and center A 

AP.AP = r\ (1) 

Conversely, if P and P 1 are two points so placed that the line 
PP passes through A and the condition (1) is satisfied, the line PP 1 
and the circle described on PP as a diameter are easily proved to 
be orthogonal to C. Then any circle through P and P is orthogonal 
to C by theorem IV, 62. Hence P and P are inverse points. 

The condition (1) shows that if one of the points P and P is 
inside of the circle, the other is outside of it. The condition holds 
also for the point A, since if AP = 0, AP =<x). By a natural 
extension of the definition of inverse points, condition (1) can also 
be taken to hold for a point on the circle (7, so that we may say 
that any point on the circle C is its own inverse. 



158 TWO-DIMENSIONAL GEOMETRY 

It is to be noticed that inverse points as here defined are also 
inverse in the sense of 53 if the circle C is a proper circle, but 
the definition given in this section is wider than that in 53, since 
it holds when the circle becomes a straight line. 

An inversion with respect to a nonspecial circle C is defined as 
a point transformation by which each point of the plane is trans 
formed into its inverse point with respect to that circle. We shall 
proceed to prove that any inversion can be represented by a linear 
transformation of tetracyclical coordinates. It is first of all to 
be noticed that by an inversion each point of the circle C is 
left unchanged by the inversion. This condition is met by the 

transformation r v ^ ^o\ 

px^^+a.^x,, (2) 

where ^ <?#= is the equation of C. Now let ]?5 t -# t -= be any 
circle through x i and its transformed point x\. Since V 6,^ = and 
V5.#J=0, we have, from (2), 

A+A+ a A+ a A=- 

If V6^.= is orthogonal to (7, we have 

(4) 



and therefore if (4) is satisfied by all values of 5< which satisfy (3), 
we may place ^~ 

dc { 

It remains to determine X. For that purpose we use the con 
dition that a) (x) = and ft) (V) = 0, and for convenience writing A 
in place of the symbol **J\c k x k , we have 

w (\x + aA)=2 \Aco (x, a) + A 2 (a) = 0. (5) 

But .() (|j) and, by (11), 63, 

($?7\ 7 N 1 7 r ^ ^ 
J = Aj?7 (c) = /? I c f- <? 
/ L i i . 

Hence co (a) = | A; [a l c 1 + 2 ^ 2 + 

i ^ -v lx-^ ^ 

and since ft) (#) = - 2^a. 

/lfm\ 

we have 



a, 



TETRACYCLICAL COORDINATES 159 

r - , . ~L^ da) k^ k . 

Therefore o> <>, a) = - 2,*,^- = g 2/^ = ^ ^, 

and, from (5), X = - w () = 77 (c). 

# 

We have consequently built up the transformation 

( 6 ) 



which is an inverse transformation, since it transforms any point x i 
into a point x\ such that any circle through x { and o^ is orthogonal 
to C. The theorem is therefore proved. It is to be noticed that the 
transformation is completely determined when the circle C is known. 
68. The linear group. We are now prepared to prove the fol 
lowing proposition : 

Any linear transformation by which the real point at infinity is 
invariant or is transformed into a point not at infinity is the product 
of an inversion and a metrical transformation. 

To prove this let T be a transformation of the form 



by means of which the relation to (x) = is transformed into itself. 
If the real point at infinity is invariant, the transformation is 
metrical ( 66). If the real point at infinity is transformed into a 
finite point A, let A be taken as the center of a circle C with respect 
to which an inversion I is carried out. By I the point A goes into 
the real point at infinity. Hence the product 7T leaves the point at 
infinity invariant and is therefore a metrical transformation. Call 
it M. Then IT = M 

whence T = 1~ 1 M=IM. 

We have written I~ l = I because an inversion repeated gives the 
identical transformation, and hence an inversion is its own inverse. 

The tetracyclical coordinates are adapted to the study of the 
properties of figures which are not altered by this group of linear 
transformations. In the geometry of these properties the straight 
line is not to be distinguished from a circle, since any point of the 
plane may be transformed into the real point at infinity, and thereby 
any circle may be transformed into a straight line and vice versa. 
Any pencil of circles may in this way be transformed into a pencil 



160 TWO-DIMENSIONAL GEOMETBY 

of straight lines and many properties of pencils of circles obtained 
from the more evident properties of pencils of straight lines. 

The distinction between special and nonspecial circles is, how 
ever, fundamental, since a circle of one of these classes is trans 
formed into a circle of the same class. 

EXERCISES 

1. Write formulas (6), 67, for the special coordinates of 57 and 
for the orthogonal coordinates of 64. 

2. From (6), 67, obtain in the coordinates of 57 the formulas for 
inversion on the circle of unit radius with its center at the origin, and 
check by changing to Cartesian coordinates. 

3. Show from (6), 67, that inversion on a fundamental circle 
of a system of orthogonal coordinates is expressed by changing the 
sign of the corresponding coordinate and leaving the other coordinates 
unchanged. 

4. Prove that a plane figure is unchanged by four inversions on 
four orthogonal circles. 

5. Show that three inversions on orthogonal circles have the same 
effect as an inversion on a fourth circle orthogonal to the three. 

6. Prove that the product of two inversions is commutative when 
and only when they take place with reference to orthogonal circles. 

7. Show that the product of two inversions on two straight lines is 
a rotation about the point of intersections of the two lines. 

8. By Ex. 7 show that the product of two inversions on the circles 
C 1 and C 2 can be replaced by the product of the inversions on two cir 
cles C[ and C z if C[ and C[ pass through an intersection of C l and C a 
and make the same angle with each other. 

9. Consider the curve denned by the quadratic equation 



Show that any circle or straight line intersects the curve in four 
points. If the coordinates are the special coordinates of 57, classify 
the curve according as (1) it does not pass through the real point at 
infinity, (2) it passes once through the real point at infinity, (3) it 
passes twice through the real point at infinity. Obtain the Cartesian 
equation for each of the classes and note the relation of the curve to 
the circular points at infinity. Note that the above classification is 
unessential from the standpoint of the linear group of tetracyclical 
transformations. 



TETRACYCLICAL COORDINATES 161 

69. Duals of tetracyclical coordinates. By anticipating a little of 
the discussion of space geometry, to be given later, we may obtain 
duals to the tetracyclical coordinates. The student to whom space 
geometry is unknown may postpone the reading of this section. 

If we interpret the ratios x l : # 2 : x 3 : # 4 as quadriplanar point 
coordinates in space of three dimensions, then 

<X)=o (i) 

is a surface of second order, and the geometry on this surface is 
dualistic with the geometry in the plane obtained by the use of 
tetracyclical coordinates. 

The linear equation ^^.= represents the plane section of 
the surface (1), and these sections are the duals of the circles in 
the plane. The point at infinity is a point on (1) not necessarily 
geometrically peculiar, and the straight lines in the tetracyclical 
plane are duals to the plane sections of (1) through this point. 

More specifically let us consider the specialized coordinates of 
57 and place in space x l : x 2 : x s : x^= z : x : y : , the usual homoge 
neous Cartesian coordinates. The fundamental equation is now 
the equation ^ + ^ __ gt = Q^ 

which, in space, represents an elliptic paraboloid. We have, then, 
the following dualistic properties: 

The tetracyclical plane The elliptic paraboloid 

The real point at infinity. The point at infinity on OZ. 

Any circle. Any plane section. 

Any proper circle. An elliptic section made by a 

plane not parallel to OZ. 
Any straight line. A parabolic section made by a 

plane parallel to OZ. 
A special circle. A section made by a tangent 

plane. 
A point circle. A section made by a tangent 

plane not parallel to OZ. 

The center of a point circle. The point of tangency. 

A special straight line. A section made by a tangent 

plane parallel to OZ (a minimum 

plane). 
The special line at infinity. The section made by the plane 

at infinity. 



162 TWO-DIMENSIONAL GEOMETRY 

Again, if we have tetracyclical coordinates for which the funda 

mental equation is 2 , , 2 , _ 

x^ -r x. 2 - 



which can be obtained from the special orthogonal system given 
in 64 by multiplying z 4 by i, the geometry obtained thereby is 
dualistic with the geometry on the surface of the sphere 



In this case the tetracyclical point at infinity is dualistic to the 
point N, where the sphere is cut by OZ. Circles on the tetracyclical 
plane are dualistic to circles on the sphere, the straight lines on 
the plane corresponding to circles through the point N on the 
sphere. This brings into clear light the absolute equivalence of a 
straight line and circle by the use of tetracyclical coordinates. In 
fact, the plane geometry on the tetracyclical plane is the stereo- 
graphic projection of the spherical geometry. 

To see this take the sphere whose equation is 



and let .2V (0, 0, 1) be a fixed point on it and P (f, 77, f) any point 
on it. The equation of the straight line NP is 

x = y = z -i j 

and this line intersects the plane z = in a point Q with the 
coordinates t n 



i-r i-r 

From these equations and the equation f 2 -f ?; 2 4- ? 2 = 1, which 
expresses the fact that P is on the sphere, we may compute 



from which, by placing 

O &> 3Un 

= -> iy = - 8 

Z 4 ^4 

we have px l x 2 + / 2 1, 



TETRACYCLICAL COORDINATES 



163 



Now, on the one hand, x l : x 2 : x a : x 4 are homogeneous Cartesian 
coordinates of a point on the sphere, and, on the other hand, they are 
tetracyclical coordinates of a point on the plane, being connected 
with the specialized coordinates of 57 by the equations 



px 1 = x{ - 






= 2 a?{, px A = x, + a? 4 , 

where x(: x! 2 : x s : x[ are the special coordinates. 

From this relation we may read off the following dualistic 

properties : 

Sphere 

Any point on the sphere. 

The point N. 

A circle (any plane section). 

A circle through N. 

A section made by a tangent 
plane. 

A section made by a tangent 
plane not passing through N. 

The point of tangency of the 
tangent plane. 

A tangent 
through N. 

A point on the plane z = 1 not 
coincident with N. 

The section made by the plane 
z = 1 (a tangent plane). 

Circles tangent to each other 
attf. 



Plane 

Any point of the plane. 

The point at infinity. 

Any circle. 

A straight line. 

A special circle. 

A point circle. 

The center of a point circle. 

A special straight line. 

The center of a special straight 
line. 

The special line at infinity. 

Parallel lines. 



plane passing 



CHAPTER X 



A SPECIAL SYSTEM OF COORDINATES 

70. The coordinate system. Each of the two coordinates x and y 
in a Cartesian system is of the type described in 7 for the coordi 
nate of a point on a line. An interesting example of a more general 
type of coordinates may be obtained by taking each of the coordi 
nates in the manner described in 8. We shall develop a little of 
the geometry obtained. The results will be of importance chiefly as 
showing that much of the ordinary 
conventions as to points at infinity 
and the ordinary classification of 
curves is dependent on the choice 
of the coordinate system. This fact 
has already come to light in the 
use of tetracyclical coordinates. The 
present chapter emphasizes the fact. 

To obtain our system of coordi 
nates take two axes OX and OY 
(Fig. 49) intersecting in at right 
angles, and on each axis take besides another point of refer 
ence, A on OX and B on OY. Then, if P is any point of the plane, 
to obtain the coordinates of P draw through P a parallel to OY 
meeting OX in M, and a parallel to OX meeting OY m N. Let the 
coordinates of M be defined as in 8 by 

k. OM x. 



N 



M 



FIG. 49 



k^AM 



and those of JVby 



/* = 




The coordinates of P may then be taken as (X, /A) or otherwise 
written as (x^. x z , y^y^)* It is clear from 8 that the ordinary 
Cartesian coordinates are a limiting case of these coordinates as A 

and B recede to infinity. 

164 



A SPECIAL SYSTEM OF COOKDINATES 165 

The coordinates being thus defined for real points the usual ex 
tension is made to imaginary points as defined by imaginary values 
of the coordinates. To consider the locus at infinity let P recede 
indefinitely from 0. This may happen in three ways : 

1. P may move on a straight line parallel to OX. Then the ratio 
x l : x 2 approaches the limiting ratio ^ : & 2 , and the ratio y l : y z has 
the constant value determined by any point on the straight line. 

2. P may move on a straight line parallel to Y. Then x 1 : x 2 has 
the constant value determined by a point on that line, and y l : y 2 
approaches the limiting value k s : k 4 . 

3. P may move on a straight line not parallel to OX or OY. 
Then M and N each approaches the point at infinity on its respec 
tive axis, and therefore the ratio x l : x z approaches ^ : k 2 and the 
ratio y l : y z approaches Jc 3 : & 4 . 

These are the only points which we recognize as at infinity. In 
other words, if P recedes indefinitely from it will not be con 
sidered as approaching a definite point at infinity unless the point 
on the curve approaches as a limit a point on a straight line. We 
have, then, the proposition i 

All points at infinity have coordinates which satisfy the equation 



To define the nature of the locus at infinity we note first that 

an equation of the type A . ON 

Oft+apiQ, (2) 

if satisfied by real points, represents a straight line parallel to OX\ 
and the equation _ 



if satisfied by real points, represents a line parallel to OY. With 
the usual extension of theorems in analytic geometry we say that 
these equations always represent lines parallel respectively to OX 
and OY. We must therefore say that equation (1) represents two 
straight lines which have the point (kj & 2 , & 3 : & 4 ) in common. We 
have, then, the proposition 

The locus at infinity consists of two straight lines having in common 
a point called the double point at infinity. 

The foregoing discussion shows that an important distinction 
between lines which are parallel either to OX or to OY and lines 



166 



TWO-DIMENSIONAL GEOMETRY 



which are not so parallel. The straight lines which are parallel to 
OX or Y we shall call special lines and divide them into two fam 
ilies of parallel lines. Lines which are not special we shall call 
ordinary lines. We have already seen that a special line has a 
point at infinity which is peculiar to itself and that all ordinary 
lines have the same point at infinity ; namely, the double point 
at infinity. We may accordingly state the following theorems, the 
proofs of which are obvious : 

/. Two special lines of the same family have no point in common. 

II. Two special lines of different families, or a special line and an 
ordinary line, have only one point in common which lies in the finite 
region of the plane. 

III. Two nonparallel ordinary lines have always the double point 
at infinity and one other finite point in common. 

IV. Two parallel ordinary lines have only the double point at 
infinity in common. 

71. The straight line and the equilateral hyperbola. From the 
equations = 



which define the coordinates, we may 
obtain 

whence OM= ^-4 



Similarly, ON= 



JL 

B 


N 1 


j 


IV 

E 


._c 


M 








O 


D\ M 


A 



FIG. 50 



Now let G (Fig. 50) be a fixed point with coordinates 
(cc^.a^ Pi-P^, let CD be the line through C parallel to OY, and 
let CE be the line through C parallel to OX. Then, if the line PM 
meets CE in M and the line PN meets CD in N , we have 



CM = OM-OD = 



CN =ON-OE= 



ak. 2 a l 



2^1 KI 



= c. 



where c l and c 2 are constants dependent upon the position of C. 



A SPECIAL SYSTEM OF COORDINATES 167 

Consider now a locus defined by the condition 

CM 



CN 



= const. 



This locus is obviously a straight line through (7, and its equation 
is of the form 



where a is a constant. 

Conversely, any equation of the form (1) in which a is not zero 

a k 8 k 
or infinity, and ^T^ T^^iT re P resen ts an ordinary straight 

#1 #1 Pi #3 
line. For (# 2 : o^, fi z :/3^) fixes a point (7, and the equation is equiva- 

, CM T , a, & 2 /3, k. 

lent to - 7= const. If a is zero, or infinity, or = -^ or 2 = - 1 , 




the equation is factorable and represents two special lines, one at 
least of which is at infinity. 

Again, consider the locus of P defined by the equation 

CM . CN = const. 

This locus is an equilateral hyperbola with two special lines as 
asymptotes. We shall call it a special hyperbola. Its equation is 



Conversely, any equation of the form (2) in which a is not zero 
or infinity, and ^y^> # ^TT represents a special hyperbola. 

^1 *1 Pi ^3 

For ( 2 : 1? ^i^) fixes a point (7, and the equation is equivalent 

a. k 8k 

to Clf (7.2V" = const. If a is zero, or infinity, or = or ^ = , 

! ^ /S, & 3 

equation (2) can be factored and represents two special lines. 

It is to be noticed that equation (1) is satisfied by the coordinates 
of the double point at infinity and that equation (2) is not. 

72. The bilinear equation. Equations (1) and (2) of 71 are of 

the form 

+ BxjJt + Cx^+ Dx 2 y 2 =0, (1) 



which is a bilinear equation in x l : x z and y l : / 2 . 

We shall now assume equation (1) and examine it in order to see 
if it is always of one of the types of 71. 



168 TWO-DIMENSIONAL GEOMETRY 

In the first place it is easy to show that the necessary and suffi 
cient condition that (1) should factor into the form 



is that ADBC=Q. Furthermore, the necessary and sufficient 
condition that (1) should be satisfied by the coordinates of the 
double point at infinity is 

Akfa+BkJc 4t + CkJc 3 +Dk 2 Jc 4 = 0. 

We shall denote the left-hand member of this equation by K and 
make four cases according to the vanishing or nonvanishing of the 
two quantities K and AD EC. 

CASE I. AD-BC=Q,K= 0. The equation cannot be factored 
and the locus does not pass through the double point at infinity. 
Therefore it cannot be of the type (1), 71. It will be of the 
form (2), 71, however, if we can find a^ # 2 , # 1? /3 2 , and a to satisfy 
the equations a&- ak&= pA, 

- (*&+ akjc 8 = P B, 

- afi^+ akfa= pC, 
afii dkjcfs pD. 

These equations can be solved by taking 



a = BC-AD. 

Hence equation (1) represents a special hyperbola. 
CASE II. AD BC 3=0,K=Q. The equation cannot be factored 
and the locus passes through the double point at infinity. We shall 
compare the equation with (1), 71. The locus of the equation 
under consideration intersects OX in the point (JDi B, 0:1), 
which we will take as (a^ : # 2 , /3 1 : /3 2 ). Using these values in (1), 
71, and comparing with (1) of this section, we have 

Bk 4 ak 2 = pA, 



A SPECIAL SYSTEM OF COORDINATES 169 



. . K ^ 

whence a * - - -> these values agreeing, since 

~ k * *i 

K= Q. Since AD BC= 0, a cannot be zero. 

Therefore the locus represents an ordinary straight line. 

CASE III. AD-BC=Q, K^O. The equation is factorable 
into the equations of two special lines, one of each family. Neither 
line can be at infinity since the locus does not pass through the 
double point at infinity. 

CASE IV. AD - BC= 0, K= 0. The equation is factorable into 
the equations of two special lines, one of each family. At least one 
of these lines must be at infinity since the locus passes through the 
double point at infinity. 

If we call a singular bilinear locus one defined by the equation (1) 
when ADI>C=Q, and a nonsingular bilinear locus one defined 
by (1) when ADBC^Q, we have the following result: 

A nonsingular bilinear locus is a special hyperbola or an ordinary 
straight line according as it does not or does pass through the double 
point at infinity. 

A singular bilinear locus consists of two special lines, one of each 
family, where one or both of the lines may be a line at infinity. 

73. The bilinear transformation. Consider the transformation 



This defines a one-to-one relation between the points (x^.x^ y^y^) 
and the points (x{ . x(, y[: y ^). The following properties are evident : 

I. Any special line is transformed into a special line of the same 
family and any singular bilinear locus into a singular bilinear locus. 

II. The lines at infinity may remain fixed or be transformed 
into any two special lines. 

III. The point at infinity may be fixed or be transformed into 
any other point either at infinity or in the finite part of the plane. 

IV. If the double point at infinity is fixed, ordinary straight 
lines are transformed into ordinary straight lines and special 
hyperbolas into special hyperbolas. 



170 TWO-DIMENSIONAL GEOMETRY 

V. If the double point at infinity is transformed into a finite 
point A and the finite point B is transformed into the double point 
at infinity, any ordinary line is transformed into a special hyperbola 
through A, and any special hyperbola through B is transformed into 
an ordinary straight line. The line AB is transformed into itself. 

EXERCISES 

1. Show that the cross ratio of the four points in which a special 
line meets four special lines of the other family is unaltered by the 
bilinear transformation. 

2. Study the transformation px[=y lt px!i = y. 2 , <ry[=x 1} a-i/ 2 = x 2) 
and also the transformation obtained as the product of this and the 
bilinear transformation of the text. 

3. Given in space the hyperboloid x*+ if z 2 1 and \ and //. defined 

by the equations 

x z 1 + y x z 1 y 

X .= = -> u, = = ^~- 

1 y x + z 1 + y x + z 

Note that (X, /-t) are coordinates of a point on the hyperboloid and 
name the essential features of a geometry on the hyperboloid which 
is dualistic to the geometry in the plane discussed in this chapter. 
Generalize by replacing the hyperboloid by any quadric surface. 

REFERENCES 

For the benefit of students who may wish to read more on the subjects 
treated in the foregoing text the following references are given. No attempt 
has been made to make the list complete or to include journal articles, 
and preference has been given to books which are easily accessible. 

General treatises : 

DARBOUX, Principes de ge ome trie analytique. Gauthier-Villars. 
KLEIN, Hohere Geometric. Lithographed Lectures. Gottingen. 
SALMON, Conic Sections. Longmans, Green & Co. 
SCOTT, Modern Analytical Geometry. The Macmillan Company. 

Projective geometry : 

EMCH, Introduction to Projective Geometry and its Applications. John Wylie & 

Sons, Inc. 

MILNE, Cross Ratio Geometry. Cambridge University Press. 
VEBLEN and YOUNG, Projective Geometry, Vol. I. Ginn and Company. 

Projective measurement and non- Euclidean geometry : 

CARSLAW, Non-Euclidean Geometry and Trigonometry. Longmans, Green Co. 
COOLIDGE, Non-Euclidean Geometry. Clarendon Press. 
MANNING, Non-Euclidean Geometry. Ginn and Company. 
WOODS, "Non-Euclidean Geometry" (in Young s Monographs on Modern 
Mathematics). Longmans, Green & Co. 



PART III. THREE-DIMENSIONAL GEOMETRY 

CHAPTER XI 

CIRCLE COORDINATES 

74. Elementary circle coordinates. As the first example of a 
geometric element determined by three coordinates, thus leading to 
a three-dimensional geometry, we will take the circle. If we con 
sider a real proper circle with the radius r and with its center at 
the point (A, &) in Cartesian coordinates, we might take the three 
quantities (A, &, r) as the coordinates of the circle. It is more 
general, however, to take the Cartesian equation 

a x (x z + y*) + a^x + a z y + a^ = (1) 

as the definition of the circle and to take the ratios a l : a 2 : a^ : a^ as 
its coordinates. The circle may then be of any of the types specified 
in 59. If it is a real proper circle the coordinates are essentially 
the same as (A, &, r). 

We may also take the equation in tetracyclical coordinates x {1 



and take the ratios u^ : u 2 : u s : u^ as the coordinates of the circle. If 
the point coordinates x i are the special coordinates of 57, the circle 
coordinates u t obtained from equation (2) are the same as the 
coordinates a. obtained from equation (1), but in general no sim 
plification is introduced by the use of the special coordinates. In 
fact, it is in many cases simpler to assume that the point coordinates 
x i in equation (2) are orthogonal. 

Unless it is otherwise explicitly stated we shall assume in the 
following that x i are orthogonal tetracyclical point coordinates 
connected by the relation : 

u(x) = xf+ xl+ xl + xl= 0. (3) 

Then the condition that equation (2) shall represent a special 
circleis ,() = + +, += 0. (4) 

171 



172 THKEE-DIMENSIONAL GEOMETRY 

As shown in 63 the equation of a special circle with the center 

& ls o> Q/, x) = y^ 4- y& 4- yfr 4- ^ =0, (5) 

where, of course, y i satisfy the fundamental relation (3). 

Hence, if (2) is a special circle the coefficients u i are exactly the 

coordinates of its center. Because of the importance of this result 

we repeat it in a theorem : 

/. If x { are orthogonal tetracyclical point coordinates and u { are circle 
coordinates based upon them, then the circle coordinates of a special 
circle are the point coordinates of the center of the circle. 

Two circles with the coordinates v i and w i are orthogonal when 

rj (y, w) = v l w 1 + v 2 w z 4 v 3 w s 4- v^w 4 =0. (6) 

From this we may deduce the following theorems : 

//. A linear equation 

a^ 4 a 2 u 2 4 aji 9 4 4 w 4 = (7) 

in circle coordinates defines a linear circle complex which is composed 
of all circles orthogonal to a base circle a l : # 2 : a g : 4 . 

For equation (7) is simply equation (6) with v i replaced by the 
constants a { and with w i replaced by the variables u { . 

The complex contains special circles whose centers are the points 
of the base circle. 

When the base circle is a. special circle the complex is called a 
special complex. It consists of all circles through the center of the 
base circle, and the condition for it is 

<-h<4- 3 2 +<=0. 

If a i are the coordinates of the real point at infinity, equation (7) 
defines a special complex consisting of all the straight lines of 
the plane. 

///. If two circles belong to a linear complex, all circles of the pencil 
defined by the two belong to the complex. 

The proof of this theorem is left to the student. 
IV. Two simultaneous linear equations 



define a linear congruence, which consists of a pencil of circles. 



CIECLE COORDINATES 173 

To prove this, note that the congruence consists of all circles 
which belong to the two complexes Va t .w.= and ^b i u i = 0. These 
circles are also common to all complexes of the pencil of complexes 

Oi+Xi,.) ,= (), (8) 

and is defined by any two complexes of this pencil. But the pencil 

(8) contains two special complexes given by the values of \ which 
satisfy the equation 

O 1 +x& 1 ) 2 +O 2 +xi s ) 2 +O 3 + A.* 3 ) 2 +O 4 +xJ 4 ) 2 = o. (9) 

If the bases of the two special complexes are distinct, the con 
gruence consists of all circles through two points and is therefore 
a pencil of circles. 

If the bases of the two special complexes coincide, equation 

(9) has equal roots. We may without loss of generality assume 
^a f Uf= to be the special complex of the pencil. Then 2aJ = 0, 
and since (9) has equal roots 2^A = ^ > that ^ s > the P om t a i is on 
the circle b { . Hence the congruence consists of all circles which 
pass through a fixed point on a circle and are orthogonal to that 
circle. They accordingly form a pencil of tangent circles. 

75. The quadratic circle complex. The equation 

2)oWb= .= a ik ) (1) 

defines a quadratic circle complex. 

Let v { and w i be any two circles. Then pu t = v i -f \w. is any circle 
of the pencil defined by v. and w i9 and belongs to the complex (1) 
when X satisfies the equation 



Hence we have the following theorem : 

/. The quadratic complex contains two distinct or coincident circles 
from any pencil of circles unless all circles of the pencil belong to the 
complex. 

Now let v { be a circle of the complex (1). Then one root of (2) 
is zero, and two roots will be zero when 

V= 0. (3) 



174 THKEE-DIMENSIONAL GEOMETRY 

Equation (3) will be satisfied by all values of w i when v i satis 
fies the equations A 



and any v i which satisfy these equations will also satisfy (1) and 
hence be the coordinates of a circle of the complex. Therefore 

//. Any circle whose coordinates v i satisfy equations (4) will be a 
circle of the complex such that any pencil of circles which contains v i 
and does not lie entirely on the complex will have only v i in common 
with the complex. 

Such a circle is called a double circle of the complex. A double 
circle does not always exist in a given complex, however, for the 
necessary and sufficient condition that equations (4) should have 
a solution is that the determinant of the coefficients should vanish. 
A complex that contains a double circle is called a singular complex. 

If in equation(2) v { is the double circle of a singular complex and 
w t any other circle of the complex, the equation is identically satis 
fied. Hence we have the following theorem : 

III. In a singular complex the pencil of circles defined by the double 
circle and any other pencil of the complex lies entirely in the complex. 

We shall now proceed to find the locus of the centers of the 
special circles of the quadratic complex. The special circles have 
coordinates u { which satisfy simultaneously equation (1) and also 
the equation for a special circle 

<+<+<+<=0. (5) 

The circle coordinates are also (theorem I, 74) the point coor 
dinates of the centers of the special circles. These coordinates 
define a one-dimensional extent. Therefore the locus of the centers 
of the special circles of the complex is a curve, which is called a 
cyclic or a bicircular curve (see Ex. 9, 68). 

The coordinates u { which satisfy simultaneously (1) and (5) will 
also satisfy the equation 

<+<+<)= (6) 



CIRCLE COORDINATES 175 

for all values of X, and any equation of the form (6) may replace 
(1) in the definition of the locus sought. But among the com 
plexes defined by (6) there are in general four singular complexes 
corresponding to the values of X defined by the equation 



Hence we have the following theorem : 

IV. The cyclic is in general the locus of the centers of the special 
circles of any one of four singular complexes. 

Take C\ any one of these singular complexes, and consider 
the straight lines belonging to the complex C. Their coordinates 
satisfy a linear equation 

W+VVt*A+W* 

where c { are the coordinates of the real point at infinity. Conse 
quently the straight lines form a one-dimensional extent, and by 
theorem I any pencil of straight lines contains two of the lines of 
this extent. Consequently the lines of the complex C envelop a 
conic, which we shall call F. 

Now let D be the double circle of (7, and T any straight line of 
C; that is, any tangent line to F. The pencil defined by D and T 
belongs entirely to (7, and consequently the two centers of the two 
point circles of this pencil are points of the cyclic. Furthermore, 
all points of the cyclic can be obtained in this way, since a point 
of the cyclic and the circle D will determine a pencil of circles 
belonging to C and containing a line T. Hence we may say : 

V. A cyclic can be defined (and in general in four ways) as the 
locus of the centers of the point circles of the pencils of circles defined 
by a fixed circle D and the tangent lines to a fixed conic F. 

Take Jj* and P^ two points on the conic F, and with P^ and P^ as 
centers construct two circles c and c orthogonal to D. The circles 
c and c determine a pencil of circles orthogonal to D and to the 
chord Pf r Hence, by theorem V, 62, if A and A are the points 
of intersection of c and c , A and A 1 are the centers of the point 
circles of the pencil of circles defined by D and the chord 



176 THKEE-DIMENSIONAL GEOMETRY 

Now let ^ approach ^ as a limit. The points A and A approach 
M and M respectively, two points on the envelope of the circles c. 
At the same time A and A approach as limits the centers of the 
point circles in the pencil of circles denned by D and the tangent 
to the conic F. Hence we have the following theorem : 

VI. A cyclic can be generated as the envelope of a family of circles 
whose centers are on a given conic T and which are orthogonal to a given 
circle D. Each circle of the family is doubly tangent to the cyclic. 

This generation of the cyclic can in general be made in four 
ways, since, as we have seen, the cyclic can be obtained from the 
point circles of four singular complexes. The cyclic curves have 
been exhaustively studied both with the use of Cartesian coordi 
nates and with the use of tetracyclical coordinates, but a further 
discussion of their properties would require too much space for 
this book. 

EXERCISES 



1. Given the equation a^u^ = 0, consider the polar equation 
a ik v { u k = 0. This assigns to any circle a definite linear complex. 

Discuss this on the analogy of polar lines with respect to a curve 
of second order in the plane, defining tangent complexes, self-polar 
systems of complexes, and the reduction of the original equation to a 
standard form. 

2. Prove that if a quadratic complex contains more than one double 
circle it contains at least a pencil of double circles and degenerates 
into two linear complexes or a single linear complex taken double. In 
the former case show that each circle of the pencil common to the two 
complexes is a double circle of the quadratic complex. 

3. If a quadratic complex degenerates into two linear complexes, 
show that the cyclic defined by it degenerates into two circles. 

4. Show that any circle in a nonsingular quadratic complex belongs 
to two pencils which lie entirely in the complex. Hence show that any 
quadratic complex is made up of two families of pencils such that any 
circle of the complex belongs to one of each of the families. Show that 
two pencils of the same families never have a circle in common and 
that any pencil of one family contains one circle of each pencil of the 
other family. 

5. Show that the following curves are special cases of cyclics : the 
ovals of Descartes, the ovals of Cassini, the cissoid, the lemniscate, 
the inverse and the pedal curves of conies. 



CIRCLE COORDINATES 177 

76. Higher circle coordinates. In addition to the four quantities 
u i> u & U 3*> u * use( i m t ne foregoing sections, we shall now introduce 
a fifth quantity w g , defined by the relation 

u * + u l + u l + ul + ul = 0. (1) 

If the point coordinates x i used in defining the elementary circle 
coordinates u { were not orthogonal, we should define u & by the 
equation > x 2 A 

7/(?/)+<=0, 

of which (1) is a special case. We may also, if we wish, replace 
the five quantities u { by five independent linear combinations of 
them, by virtue of which equation (1) would be transformed into 
a more general quadratic equation, so that we may say the higher 
circle coordinates in their most general form consist of the ratios of 
Jive variables connected by a fundamental quadratic relation 



We shall continue to use the orthogonal form for simplicity of 
treatment. 

As shown in 59 the vanishing of the coordinate u 5 is the neces 
sary and sufficient condition that the circle should be special. In 
this case the circle is completely determined by the four coordi 
nates u^ u z , te a , u 4 . So, in general, the center and the radius of a 
circle are fully determined by means of the first four coordinates, 
u i> u i"> u tf u t> that is, the circle is completely determined in the 
elementary sense. The absolute value of u 5 is then determined, but 
its sign is not fixed. 

It is necessary, then, to distinguish between two circles which are 
alike in the elementary sense but differ in the sign of the coordi 
nate u b . This may be done by noting that any nonspecial circle, 
whether a proper circle or a straight line, divides the plane into two 
portions, and by considering a circle with a fixed u b as the boundary 
of one of these portions and the circle with a coordinate u of 
opposite sign as the boundary of the other portion. The same result 
may be obtained by considering the circle described in opposite 
directions, with the agreement, perhaps, that the circle shall be 
considered as bounding that portion of the plane which lies on the 
left hand in describing the circle. 



178 THREE-DIMENSIONAL GEOMETRY 

If x { are the orthogonal coordinates described in detail in 64, 
that is, if we introduce Cartesian coordinates so that 



it is easy to compute that the radius of the circle u { is equal to 
Hence to fix a sign of U B is equivalent to fixing the sign 

of the radius. We may agree that the sign of the radius is to be 
considered positive when the center of the circle lies in the area 
bounded by the circle and that the sign of the radius is to be 
taken as negative when the center lies in the part of the plane not 
bounded by the circle. 

The angle between two circles u { and v t . is now defined without 
ambiguity by the formula 

- u. v + u.v + uv + u.v 
cos 9 = - - - *-* 

"& 

or w^ 4- u z v z 4- u s v s + u^v^ + u 5 v 5 cos 6 0. (2) 

To change the sign of u b but not of v & is to change the angle 
into its supplementary angle. 

If the circles u { and v. are real and the coordinates are those of 
64, it is not difficult to see that the angle 6 is the angle between 
the two normals drawn each into the region of the plane which 
each circle bounds. 

If either of the two circles is special, 6 is either infinite or in- 
determinant. In particular, if v i is a special circle and u { is not, 
we have cos = oo when the center of v i does not lie on u^ and 

cos = - when the center of v i lies on w.. Hence we may say : 
A special circle makes any angle with a circle on which its center lies. 

Two circles are orthogonal when 6 = (2 & + !) The necessary 
and sufficient condition for this is 

u^ + u 2 v 2 + u s v 3 + uji^ =0. (3) 

Two circles are tangent when 6 = 0. The necessary and sufficient 
condition for this is 



It is to be noted that two circles are not defined as tangent when 
6 = TT. If the circles are real proper circles they are tangent only 



CIRCLE COORDINATES 179 

when they are tangent in the elementary sense and the interior of 
one lies in the interior of the other. 
Consider the equation 



in the higher circle coordinates. This is equivalent to equation (2) 
if we place 

a i = V *= v a z= V B a *= v s 5 =v 5 cos<9, 
together with the condition 

tf+t^+^+^+^O. 

These equations are just sufficient to determine v i and cos 6. Hence 
the higher circle complex consists of circles cutting a fixed circle under 
a fixed angle. 

If a & = the higher circle complex becomes the elementary com 
plex consisting of circles orthogonal to a base circle. 

The circle complex (5) is called a special complex when 

<4X+tf 3 2 +<+<=0. 

In that case 0=^0 and the equation may be identified with (4). 
Hence a special complex in the higher coordinates consists of circles 
tangent to a fixed circle. 

Two simultaneous equations 

a^+ a 2 u 2 + 3 w 3 + a^u 4 + a & u 6 = 0, 

l l U l + b 2 U 2 + b S U 3 + 6 4 W 4 + ^5 = 

define a higher circle congruence. Circles which satisfy these two 
equations also satisfy any equation of the form 



but among the complexes defined by this last equation are two 
special complexes. Hence a higher circle congruence consists of all 
circles tangent to two fixed circles. 

EXERCISES 

1. What is the configuration of the higher circle congruence if the 
two special complexes coincide ? 

2. Show that if x { are orthogonal tetracyclical coordinates, the circle 
coordinates u lt u^ u s , u 4 are proportional to the cosines of the angles 
which the circle u f makes with the coordinate circles. 

3. Describe the complexes defined by each of the equations u { = 0. 



CHAPTER XII 



POINT AND PLANE COORDINATES 

77. Cartesian point coordinates. Let OX, OF, OZ (Fig. 51) be 
three axes of coordinates, which we take for convenience as mutu 
ally orthogonal. Then, if P is any point in space, and PL, PM, 
PN are the perpendiculars to the three 
planes determined by the axes, the 
lengths of these perpendiculars with a 
proper convention as to signs are the 
rectangular Cartesian coordinates of P. 
That is, we place 



N 



= LP, z = 



(1) 




FIG. 51 



where MP, LP, and NP are positive if 
measured in the directions OX, F, and 
OZ respectively, and negative if measured in the opposite directions. 
The coordinates may be made homogeneous by placing 



MP = -, 
t 



(2) 



and taking the ratios xiy.z-.t as the coordinates of P. 

To any point P corresponds then a real set of ratios, and to any 
set of real ratios in which t is not zero corresponds a real point P. 
The relation between point and coordinates is then made one to 
one by the following conventions : (1) the ratios 0:0:0:0 are 
not allowable ; (2) complex values of the ratios define an imag 
inary point; (3) ratios in which = but x\y\z are determinate 
define a point at infinity. In fact, as t approaches zero P recedes 
indefinitely from 0. 

If a point is not at infinity we may, if we choose, place = 1 
in (2), thus reducing the homogeneous coordinates to the non- 
homogeneous ones. Again, nonhomogeneous coordinates are easily 
made homogeneous by dividing by t. Accordingly we shall use 

180 



POINT AND PLANE COOKDINATES 181 

the two kinds side by side, passing from one to the other as 
convenience dictates. 

A more general system of Cartesian coordinates may be defined 
by dropping the assumption that the axes OX, OY, OZ (Fig. 51) 
are mutually orthogonal, and drawing the lines HP, LP, NP 
parallel to the axes. The coordinates are then called oblique. They 
may be made homogeneous by the same device as that used in the 
case of rectangular coordinates. 

Throughout this book the axes will be assumed as rectangular 
unless the contrary is explicitly stated. 

78. Distance. Let J^ and P^ be two real points with the coordi 
nates (Xf y^ z^) and (z , y^ 2 2 ) respectively, and let a rectangular 
parallelepiped be constructed on P^ as a diagonal, with its edges 
parallel to the coordinate axes. Then, if PJi, RS, and SP Z are three 
consecutive edges of the parallelepiped, it is evident that 

P l R = x 2 -x 1 , SS = y t -y p -WJ= *,-*,. (1) 

Hence the distance P^ is given by the equation 

yO +c*,-*,) . (2) 



of, written in homogeneous coordinates, 



+ OA - * 



This formula has been proved for real points only. It is now 
taken as the definition of the distance between all points of what 
ever nature. From the definition we obtain at once the following 
propositions : 

I. The distance between two points neither of which is at infinity is 
finite. 

II. TJie distance between a point at infinity and a point not at infinity 
is infinite, unless the point at infinity has coordinates which satisfy 
the conditions 



2 2 A A 

+2 = 0, = 0. (4) 

In the latter case the distance between the point at infinity and any 
point not at infinity is indeterminate. 

The points whose coordinates satisfy equations (4) form a one- 
dimensional extent called the circle at infinity. The reason for the 
use of the word "circle" will appear later. 



182 THREE-DIMENSIONAL GEOMETRY 

If in equation (2) we replace the coordinates of P l by those of a 
fixed point C (X Q , y^ z ) and the coordinates of P l by those of a 
variable point P (#, ?/, z), while keeping CP equal to a constant r, 
we obtain 



(a . _ x 

which defines the locus of a point at a constant distance from a 
fixed point. This locus is by definition a sphere. 
Equation (5) may be written in the form 

= 0, (6) 



where 

VVVi-jr:*,*,-^ ^+^-4^. (7) 

If the center C and the radius r are finite, the coefficient A is not 
zero. Conversely, any equation of the form (6) in which A is not 
zero defines a sphere, the radius and the center of which are given 
by (7). More generally it is possible to define a sphere as the 
locus of any equation of the form (6). In case A= the center is 
at infinity, the radius is infinite or indeterminate, and the equa 
tion splits into the two equations t = and Bx + Cy +Dz + Et = 0. 
These cases of the sphere will be discussed in detail in 118. In the 
present section we shall consider only the case in which A= and 
the sphere conforms more nearly to the elementary definition, and 
its equation may then be put in the form (5). 

The radius, however, may be real, imaginary, or zero. If the 
radius is zero, the equation takes the form 

(?-*.) + (y-y t y+(?-tf=0, (8) 

and the sphere is called a null sphere or a point sphere. 

It is obvious that if (X Q , y Q , ) is a real point, equation (8) is 
satisfied by the coordinates of no other real point. There exist, 
however, a doubly infinite set of imaginary points which satisfy 
equation (8). 

79. The straight line. A straight line is by definition the one- 
dimensional extent of points whose coordinates satisfy equations 
of the form pz = Xl +\x^ 

PV = &!+*!/* (1) 

/>Z = J2 t + \Z 2 , 

pt = ^ + X 2 , 



POINT AND PLANE COORDINATES 183 

where (^ : y l : z 1 : t^ and (x 2 : y z : z z : Q are the coordinates of two 
fixed points and X is a variable parameter. 

From the definition we may draw the following conclusions : 

/. Any two distinct points determine a straight line, and any two 
distinct points on the line may be used to determine it. 

The first part of this theorem is obvious. To prove the second 
part let P^ be a point on the line (1) determined by X = \ l and let 
P 2 be another point on the line determined by X = X 2 . Let a- be 

a quantity defined by the relation ^ 2 = X. Then the first 

equation in (1) may be written 

px = Xl 

or TZ = X I 

and similar equations can be found for y, z, and t. But these are 
the equations of a straight line defined by ^ and ZJ, which is 
thus shown to be identical to that defined by (x^i y^z^: t^) and 

Cv# 2 : vO- 

//. A straight line contains a single point at infinity unless it lies 
entirely at infinity. 

If, in equations (1), ^=0 and t 2 = 0, then t = f or all values of X. 
Otherwise t = only when X = -> which determines on the line 

^2 

the single point at infinity (xfa- x&i yf^- yfo zf z - z^i 0). This 
proves the theorem. Straight lines which lie at infinity are some 
times called improper straight lines; other lines are called proper 
straight lines. 

III. If two points of a straight line are real, the line contains an 
infinity of real points. 

This follows from the fact that if the two real points are used to 
determine the equations (1), any real value of X gives a real point 
on the line. Such lines are called real lines, although it should not 
be forgotten that they contain an infinity of imaginary points also. 

If a real line is also a proper line we may put t^ t <2 , and t equal 
to unity in equations (1) and write the equations of the line in 
the form 

x ~ x 



184 THREE-DIMENSIONAL GEOMETRY 

From this and equations (1), 78, it is not difficult to show 
that the real points of a real proper line form a straight line in the 
elementary sense. 

IV. An imaginary straight line may contain one real point or no 
real point. 

To prove this it is only necessary to give an example of each 
kind. The line defined by the two points (1:1:1:1) and (1:0: i:l) 
contains the first point and no other real point, while the line 
defined by (\:i:i: 1) and (1:0:^:1) contains no real point. 
These statements may be verified by using the given points in 
equations (1) and examining the values of X necessary to give a 
real point on the line. 

An imaginary line which contains no real point may be called 
completely imaginary, one with a single real point incompletely 
imaginary. 

V. If the distance between two points on a straight line is zero, the 
distance between any other two points of the line is zero. 

To prove this we may use the coordinates of the points between 
which the distance is zero for the fixed points in equation (1). 
Then, if ^ and P z are two points determined by X = \ and \ = \ 
respectively, we may compute the distance P^P? by formula (3), 
78. There results 



A straight line with the above property is called a minimum line. 
Such lines have already been met in the plane geometry. Concern 
ing the minimum lines in space we have the following theorems: 

VI. A minimum line meets the plane at infinity in the circle at 
infinity, and, conversely, any line not at infinity which intersects the 
circle at infinity is a minimum line. 

From the proof of theorem II the necessary and sufficient con 
dition that a line meet the circle at infinity is 

(*A - *A>*+ frA- yA> + OA- .*,) = . 

which is also the necessary and sufficient condition that the two 
points (xjyjzjt^) and (x z y^zjt^) should be at a zero distance 
apart. By theorem V the line is then a minimum line. 



POINT AND PLANE COORDINATES 185 

VII. Through any point of space goes a cone of minimum lines 
which is also a point sphere. 

Any point in space may be joined to the points of the circle at 
infinity. We have then a one-dimensional extent of lines through 
a common point, and such lines form a cone by definition. Also 
if (X Q : y Q : Z Q : t Q ) is the fixed point and (x\y:z\ f) is any point on a 
minimum line through it, the coordinates of (x : y : z : ) will satisfy 
the equation 



^ 2 ^2 A 

- xf) 2 +O - y t) f -h(^ - z = 0, (3) 

and, conversely, any point whose coordinates satisfy this equation 
lies by theorem VI on a minimum line through (X Q : y Q : Z Q : Q ). 

Equation (3) is, however, the equation of a point sphere in 
homogeneous form. Hence the minimum cone is identical with 
the point sphere. 

80. The plane. A plane is defined as the two-dimensional extent 
of points whose coordinates satisfy an equation of the form 

Ax + y+Cz + Dt = Q. (1) 

From the definition we deduce the following propositions : 

/. If two points lie on a plane, the straight line connecting them lies 
entirely on the plane. 

This follows immediately from the fact that if (xj y^.z^. ^) and 

fe^VO satisf y (1) tnen (^+H : #i+ X #2 : 2i+ X V*i+ X Q 
does also. 

//. A plane is uniquely determined ~by any three points not on the 
same straight line. 

If (xjyjzjt^ 0*v y a : V *a) and (xjyjzjt^ are any three 
points, the coefficients A, B, (7, and D may be so determined that 

= 0, 

= 0, (2) 

< 8 = 0, 
unless there exist relations of the form 



that is, unless the three points taken lie on a straight line. 



186 THREE-DIMENSIONAL GEOMETRY 

It follows from theorems I and II that any plane in the elemen 
tary sense may be represented by an equation in the form (1). 
The general definition of a plane extends the concept of the plane 
in the usual way. 

777. Points at infinity lie in a plane called the plane at infinity. 

This is a result of the definition, since the equation of points at 
infinity is t = 0. 

On the plane x = the coordinates y:z:t are homogeneous 
coordinates of the type of 18. Similarly, on the plane y = we 
have the Cartesian coordinates xizit and on the plane z = the 
Cartesian coordinates x:y:t. On the plane t = we may define 
x:y:z as trilinear coordinates of the type in 22. 

IV. If three points of a plane are real, the plane contains a doubly 
infinite number of real points. 

From equations (2) the values of A, B, C, and Z> are real if the 
coordinates of the points involved are real. Then in equations (1) 
real values may be assumed for two of the ratios xiy:z:t, and the 
third is determined as real. 

Such a plane is called a real plane, although it contains, of course, 
an infinity of imaginary points. 

V. Any two distinct planes intersect in a straight line, and any 
straight line may be defined as the intersection of two planes. 

Consider the two planes 

Ap + By + Op + DJ = 0, 



These equations are satisfied by an infinite number of values of 
the coordinates. Let (xj yjzj ^) and (ay / 2 : z 2 : 2 ) be two such 
values. Then the values (x t ^-\se^:y 1 ^-\y t :z l ^-\f t :t 1 +\t^) also 
satisfy the two equations so that the two planes have certainly a 
line in common. They cannot have in common any point not on 
this line if the two planes are distinct, since three points completely 
determine a plane (theorem II). 

Again, a plane (by theorem II) may be passed through two points 
on a given line and a third point not on the line, and two such 
planes will determine the line. 



POINT AND PLANE COORDINATES 187 

VI. Any plane except the plane at infinity contains a single line at 
infinity, and any two planes intersecting in the same line at infinity 
are parallel. 

The first part of this theorem is a corollary of theorem V. The 
second part is a definition of parallel planes. The definition agrees 
with the elementary definition since, by theorem V, parallel planes 
in this sense have no finite point in common. 

VII. An imaginary plane contains one and only one real straight line. 

Since an imaginary plane has one or more of the coefficients in 
its equation complex, we may write the equations as 

* + S it = 0. 



This can be satisfied by real values (x\y:z\i) when and only 
when 



that is, when (x: y.zit) lie on a real straight line (theorem V). 
That the line is real follows from theorem III, 79, since the above 
equations are evidently satisfied by two real points. 

The real line on an imaginary plane may lie at infinity. In 
that case the plane is said to be imaginary of higher order. If the 
real line is not at infinity, the plane is said to be imaginary of 
lower order. 

VIII. Any plane intersects a sphere in a circle. 
Consider the intersection of the plane 

Ax + By + Cz + D = (3) 

and the sphere 

a(x* + y*+ z 2 ) + lx + cy + dz + et = Q. (4) 

Any point on the intersection of these two surfaces also lies on 
the intersection of (3) and 

a(a* + y a + 3 2 ) + (6 + \A)x + (c + \B)y + (c? + \CT)z 

+ (e + X>)*=0, (5) 

where X is any multiplier. Equation (5) represents a sphere with 



which will lie in the plane (3) when 
bA + cB + dC 



188 THKEE-DIMENSIONAL GEOMETRY 

The points of the intersection of (3) and (4) are therefore 
shown to lie at a constant distance from a fixed point of the 
plane, and hence the intersection satisfies the usual definition of 
the circle. 

The above discussion fails if the coefficients of the plane satisfy 
the condition ^ 2 _j_# 2 _f_ c 2 = 0. 



This happens for the plane at infinity and for other planes called 
minimum planes. In these two cases the truth of theorem VIII is 
maintained by taking it as the definition of a circle. This justifies 
the expression "circle at infinity," which we have already used, 
and shows that there is no other circle at infinity. The case of a 
minimum plane needs further discussion. 

IX. Any plane not a minimum plane intersects the circle at infinity 
in two points, which are the circle points of that plane. A minimum 
plane is tangent to the circle at infinity. Through any point in a plane 
which is not a minimum plane go two minimum lines. Through any 
point in a minimum plane goes only one minimum line. 

The plane (3) intersects the plane at infinity in the line 
Ax 4- By -f Cz = 0, t = 0, and this line intersects the circle at infinity 
in two points unless A 2 +B 2 + (7 2 = 0, when it is tangent to that circle. 
In the latter case the plane is by definition a minimum plane. 

It is easy to see that in a plane which is not a minimum plane 
its intersections with the circle at infinity have all the properties of 
the circle points discussed in 20 and that the metrical geometry 
on such a plane is that of 45 and 46. The latter parts of the 
theorem follow from theorem VI, 79. 

The minimum planes are fundamentally different from other 
planes in that a minimum plane contains only one circle point at 
infinity. The geometry on a minimum plane presents, therefore, 
many peculiarities, some of which will be mentioned in the next 
section. 

81. Direction and angle. We define the direction of a straight 
line as the coordinates of the point in which it meets the plane at 
infinity. This definition is justified by the facts that the lines 
through a point are distinguished one from another by their direction 
in accordance with theorem I, 79, and that a line can be drawn 
through the point with any given direction by the same theorem. 



POINT AND PLANE COORDINATES 189 

We shall denote the direction of a line by the ratios l:mfn. 
Then we have, by theorem II, 79, 



where (x^ : y l : z l : ^) and (# 2 : y z : z 2 : t^) are the coordinates of any 
two points of the line. If neither of these points is at infinity, we 

may write , r . 7 , _ 7 , . ~ _ ~ 

i . m . n x z x l . y z y^ . z z z^ 

which is in accordance with the more elementary definition of 
direction. 

From the definition we have the following consequences : 

/. Two noncoincident lines with the same direction are parallel. 

Such lines lie in the plane determined by their common point at 
infinity and two distinct points one on each line (theorem II, 80), 
and they can intersect at no point except the common point at 
infinity. Hence they are parallel. 

//. The necessary and sufficient condition that a line should be a 
minimum line is that its direction should satisfy the condition 



This follows from (3), 79. 

In 46 we have defined the angle between two intersecting lines 
j and l z by the equation 



where m^ and w 2 are the two minimum lines through the inter 
section of ^ and 1 2 and in their plane. We shall continue to use 
this definition. 

Now, if the lines I l9 Z , m^ and m 2 intersect the plane at infinity in 
the points L^ L^ M^ and M 2 respectively, we have, by theorem I, 16, 



From this we have the following theorem, in which the condition 
that ? t and ? 2 should be intersecting lines may be dropped : 

///. The angle between two lines is equal to the projective distance 
between the points in which they intersect the plane at infinity, the 
circle at infinity being taken as the fundamental conic and the constant l 

0/(4), 47, being equal to - 



190 THREE-DIMENSIONAL GEOMETRY 



"The cross ratio (L^L^ M^f^) is unity when and only when M^ 
and M 2 coincide or L^ and L 2 coincide, it being assumed that neither 
L^ nor L 2 lies on the circle at infinity. In the former case the lines 
Zj and Z 2 are parallel ; in the latter case they lie in the same minimum 
plane. Hence follows the theorem : 

IV. If two nonminimum lines are parallel or if they lie in the same 
minimum plane, they make a zero angle with each other, and, con 
versely, if two nonminimum lines make a zero angle with each other, 
they are either parallel or lie in the same minimum plane. 

Let us suppose that Z x and 1 2 are nonminimum and distinct and 
that their directions are A^iBj C^ and A 2 :B 2 : C 2 respectively. Then, 
as in (4), 49, 



* + A a + c? ^ 2 2 + BI 4- ci 

From this we obtain the following result: 

V. Two nonminimum lines are perpendicular to each other when 
their directions satisfy the condition 

A l A t + B 1 B 1 + 0,07,= 0. (2) 

Interpreted on the plane at infinity this means that the two 
points (A^Bj (7 1 ) and (A 2 :B 2 i (7 2 ) lie each on the polar of the other. 

VI. If Ax 4- By 4- Cz + Dt = is not a minimum plane, any line 
with the direction A : B : C does not lie in the plane and is perpen 
dicular to every line in the plane. 

The plane mentioned meets the plane at infinity in the line 
Ax 4- By 4- Cz = 0, and any line with the direction A: B: C meets 
the plane at infinity in the point (A : B : (7), which is the pole of the 
line Ax 4- By 4- Cz with respect to the circle at infinity. Hence 
the point (A:B:C) will not lie in the line Ax+By -f Cz = unless the 
latter is tangent to the circle at infinity. This proves the theorem. 

Any line with the direction A : B : C is said to be normal to the 
plane Ax+By+Cz+Dt= 0, and this designation is used sometimes 
even for minimum planes. The above discussion, however, estab 
lishes the following theorem: 

VII. The normals to a minimum plane lie in the plane and are the 
minimum lines in the plane. 



POINT AND PLANE COORDINATES 191 

By (1) a line with the direction l:m:n makes with the axes of 
coordinates the angles a, /8, 7, where 

/ n m *> 

png n. = > COSp= =1 COS 7 = 



These quantities are called the direction cosines of the line. 
With their use equations (2) of 79 may be put in the form 

x = x l -\-r cos a, 



z = z 1 +r cos 7, 

where it is easy to show that r is the distance of the variable point 
(x, y, z) from the fixed point (x^ y v , z^). It is obvious that these 
equations do not hold for a minimum line. 

EXERCISES 

1. Show that through any imaginary point in space there goes a 
pencil of real planes having a real line as axis. 

2. Show that the equation of any imaginary plane of lower order 
may be written ax -f- by -f- cz -f- dt = 0, where a, b, and c are real and d 
is complex. 

3. Show that any imaginary straight line either lies in one real 
plane and contains one real point, or lies in no real plane and contains 
no real point. The last kind of lines is called completely imaginary 
and the former kind incompletely imaginary. 

4. Show that the necessary and sufficient condition that two points 
should determine an incompletely imaginary straight line is that the 
two points lie in the same plane with their conjugate imaginary points, 
but not on the same straight line. 

5. Show that two conjugate imaginary points determine a real 
straight line and that if an imaginary point lies on a real straight line 
its conjugate imaginary point does also. 

6. Show that a minimum line makes an infinite angle with any 
other line not in the same minimum plane with it and makes an inde 
terminate angle with any line in the same minimum plane with it. 

7. If (2) is taken as the definition of perpendicular lines, show that 
a minimum line is perpendicular to itself and that a line in a minimum 
plane is perpendicular to every minimum line in the plane. 



192 THREE-DIMENSIONAL GEOMETRY 



*. If the angle between two planes is the angle between their 
normals, show that two nonminimum planes make a zero angle when 
they are parallel or intersect in a minimum line. 

9. Show that any minimum plane makes an infinite angle with any 
/plane not intersecting it in a minimum line and makes an indeterminate 
angle with any plane intersecting it in a minimum line. 

10. Show that the coordinates of a point on the circle at infinity 
can be written x : y : z = 1 s 2 : i (1 +- s 2 ) : 2 s, where s is an arbitrary 
parameter. Hence show that the equations of a minimum line may be 

written . /.. ON 

- 



where s is fixed for the line and r is variable. 
11. Show that the equations 

x= C(l- 



where F(s) is an arbitrary function, represent a minimum curve; that 
is, a curve such that the length between any two points is zero and 
the tangent line at any point is a minimum line. 

12. Show that a minimum plane through the center of a sphere 
intersects the latter in two minimum lines intersecting at infinity. 

13. If a line is defined by the two equations 

Ap + By + Cp + Df = 0, 
A 2 x -f- B$ 4- Cfi 4- Dcf = 0, 
show that its direction is B^C^BjO^iC^A.^ C^A^.Af^ A 2 B r 

14. Show by reference to the plane at infinity that the necessary 
and sufficient condition that the plane Ax + By + Cz + Dt should 
be parallel to a line with direction I : m : n is A I -(- Bm + Cn = 0. 

15. Show that the equation of a plane through the point (x^. y^ z^ ^) 
and parallel to the two lines with the directions l^\ m^\ n^ and 1 2 : m 2 : n^ 

respectively, is 

y z t 

y\ ^i ^i 

m l n l 
m n n n 



POINT AND PLANE COORDINATES j 

82. Quadriplanar point coordinates. Let us assume four planes of 
reference ABC, ABD, ADC, and BCD (Fig. 52), not intersecting in 
a point, and four arbitrary constants k^ Jc 2 , k s , k 4 . Let p^ p 2 , p s , p^ 
be the lengths of the perpendiculars from any point P to the four 
planes in the order named, the sign of each perpendicular being 
positive or negative according as P lies on one or the other (arbi 
trarily chosen) side of the corresponding plane. Then the ratios 




are the coordinates of the point P. 

It is evident that if P is given as a real point its coordinates are 
uniquely determined. Conversely, let a set of real ratios x^ixj x & : x^ 
be given, no one of which is zero. The 
ratio x 1 : # 4 is one of the coordinates of 
any point in a definite plane through 
BC, and the ratio x 2 : # 4 is one of the 
coordinates of any point on a definite 
plane through BD. The two ratios are 
part of the coordinates of any point on a 
definite line through B and of no point 
not on this line. Call this line I. The 
ratio x z : x^ is one of the coordinates of FIG 52 

any point on a definite plane through 

CD. Call this plane m. If the plane m and the line I meet in a 
point P, the ratios x 1 : x 2 : x s : x^ have fixed a definite point. If the 
line I and the plane m do not intersect, we shall say that the ratios 
define a point at infinity. 

Complex values of the ratios define imaginary points, and the 
ratios 0:0:0:0 are excluded. 

If one of the coordinates is zero, the other three are trilinear 
coordinates on one of the planes of reference. For example, if x l = 
the ratios x 2 : x^: x^ are trilinear coordinates in the plane ABC, since 
the distance of a point in the plane ABC from the line AC is equal 
to its distance from the plane ACD multiplied by the cosecant of 
the angle between the planes ABC and ABD, and, similarly, for the 
distances from AB and BC. 

Hence all values of the ratios x l : x 2 : x 8 : # 4 , except the unallow 
able ratios 0:0:0:0, determine a unique point. 



194 THREE-DIMENSIONAL GEOMETRY 

Referring to the figure, we note that x t = on the plane ABC\ 
x z = on the plane ABD; x a = on the plane ADC; and x^= 
on the plane DEC. 

The point A has the coordinates 0:0:0:1, the point B the 
coordinates 0:0:1:0, the point C the coordinates 0:1:0:0, the 
point D the coordinates 1:0:0:0. The ratios k 1 : Jc 2 : Jc B : & 4 are 
determined by the position of the point /, for which the coordinates 
are 1:1:1:1, and this point can be taken at pleasure. 

Quadriplanar coordinates include Cartesian coordinates as a spe 
cial or limiting case in which the plane x^= is taken as the plane 
at infinity. For if the plane BCD recedes indefinitely from A, and 
the point P is not in BCD, the perpendicular p^ becomes infinite in 
length, but & 4 can be made to approach zero at the same time and 
in such a manner that Iim&j0 4 = l. Finally, if the planes ABC, 
ABD, and ACD are mutually orthogonal and ^ l= =^ 2 =^ g = l, the 
coordinates are rectangular Cartesian coordinates. 

If the planes ABC, ABD, and ACD are not mutually orthogonal, 
we may place & x = esc a^, where a 1 is the angle between AB and the 
plane ACD, and take similar values for Jc 2 and Jc s . We then have 
oblique Cartesian coordinates. 

In using quadriplanar coordinates it is not convenient or neces 
sary to specify the coordinates of a point at infinity. In fact, such 
points are not to be considered as essentially different from other 
points. Distance and all metrical properties of figures are not 
conveniently expressed in terms of quadriplanar coordinates and 
should be handled by Cartesian coordinates. We may, however, 
pass from the general quadriplanar coordinates to Cartesian coordi 
nates by simply interpreting one of the coordinate planes as the 
plane at infinity. 

83. Straight line and plane. We shall prove the following theorems : 

/. If y^ : ?/ 2 : y z : y^ and z l : z 2 : Z Q : z^ are two fixed points, the coordi 
nates of any point on the straight line joining them are 



and any point with these coordinates lies on that line. 



POINT AND PLANE COORDINATES 195 

This is the definition of a straight line for imaginary points. If, 
however, the points y i and z i are real, the points given by real 
values of X are real points which lie on a real straight line in the 
elementary sense. This is easily verified by the student in using 
a construction and argument similar to that used in 23 for the 
straight line in the plane. 

II. Any homogeneous linear equation of the form 



represents a plane. 

This is the definition of a plane. If y and z i are any two points 
satisfying the equation of a plane, the coordinates of any point on 
the line joining y. and z. also satisfy the equation ; that is, the line 
which joins any two points of a plane lies entirely in the plane. 
Hence, if the plane contains real points it coincides with a plane 
in the elementary sense. 

///. Three points not in the same straight line determine one and 
only one plane. 

The proof is as in 80. If y^ z { , t { are the three points, the 
equation of the plane is 



y, 



= 0. (3) 



IV. If y { , z i9 and t t are any three points not on the same straight 
line, the coordinates of any point on the plane through them may be 
written 



and any point with these coordinates lies in the plane. 

This follows immediately from the fact that the elimination of 
p, X, and IJL from equations (4) gives equation (3), and, conversely, 
from (3) the existence of (4) may be deduced. 



196 



THREE-DIMENSIONAL GEOMETRY 



V. Any two distinct planes intersect in a straight line. 

The proof is the same as that of theorem V, 80. A line can 
therefore be denned by two simultaneous equations of the form 



VI. If ^jd^i and *&&= are the equations of any two 
planes, then 



is, for any value of \, the equation of a plane through the line of in 
tersection of the first two planes. As \ takes all values, all planes of 
the pencil may be obtained. 

VII. Any three planes not belonging to the same pencil intersect in 
a point. 

To prove this consider the three equations 



These have the unique solution 



unless the determinants involved are all zero. But in the latter 
case there must exist multipliers X, /-i, p such that 



and hence the three planes belong to the same pencil by theorem VI. 

VIII. If ^a { x.= 0, V&^= 0, V^f = ^ are ^e equations of three 
planes not belonging to the same pencil, then 



Z8 the equation of a plane through their point of intersection. As X 
and JJL take all values, all planes through a common point can be found. 
Such planes form a bundle. 

The proof is obvious. 



POINT AND PLANE COOKDINATES 197 

84. Plane coordinates. The ratios of the coefficients in the equa 
tion of the plane are sufficient to fix the plane and may be taken 
as the coordinates of the plane. We shall deno te them by u { and say 
that u^u z :u z : u^ are the plane coordinates of the plane whose point 
equation is u ^ u ^ uf ^ Uf _ . (1) 



No difference is made in this definition if the point coordinates 
are Cartesian. Equation (1) is the condition that the plane u t and 
the point x { should be in united position ; that is, that the plane 
should pass through the point or that the point should lie on the 
plane. 

We have the following theorems, which are readily proved by 
means of those of 83 : 

/. If vj v^: v s : v^ and w^.w^.w^. w^ are the coordinates of two fixed 
planes^ the coordinates of any plane through their line of intersection are 






and any plane with these coordinates passes through this line. 

The proof is obvious. Equations (2) are the equations of a 
pencil of planes. They are also called the plane equations of a 
straight line, the axis of the pencil. In this method of speaking 
the straight line is thought of as carrying the planes of the 
pencil in the same sense as that in which by the use of equa 
tions (1), 83, the straight line is thought of as carrying the 
points of a range. 

//. Any homogeneous linear equation of the form 

a , u , + a * u z + <y* 8 + a^ = (3) 

is satisfied by the coordinates of all planes through a fixed point. 

It follows from (1) that all planes whose coordinates satisfy (3) 
are united with the point a l : a 2 : a & : a^. Equation (3) is therefore 
called the plane equation of the point a l : a 2 : a g : 4 , in the same 
sense in which equation (2), 83, is the point equation of the 
plane a l : a 2 : a s : 4 . 



198 THREE-DIMENSIONAL GEOMETRY 

///. Three planes not belonging to the same pencil determine a point. 

This is, of course, the same theorem as VII, 83, but in plane 
coordinates we prove it by noticing that three values of u { , say v { , 
w.j s i9 which satisfy (3) are sufficient to determine the coefficients 
of (3) unless p8 t =\Vf-\- fiw^ The equation of the point determined 
by the three planes is, then, 



1 U 2 U t 



V 3 \ 



w 2 w s w 4 



= 0. (4) 



IV. If v i9 Wf, and s { are any three planes not "belonging to the same 
pencil, the coordinates of any plane through their common point are 

pU. V t + \Wi+ fJL8 f , 

and any plane with these coordinates passes through this point. 
The proof is obvious. These planes form a bundle. 

V. Two linear equations which are distinct are satisfied by the coordi 
nates of planes which pass through a straight line. 

This follows from the fact that each equation is satisfied by 
planes which pass through a fixed point. Simultaneously, therefore, 
the equations are satisfied by planes which have two points in com 
mon, and these points are distinct if the equations are distinct. The 
planes, therefore, have in common the line connecting the two points. 

The equation of a straight line can therefore be written in 
plane coordinates as the two simultaneous equations 

Vl+ a 2 W 2+ V 3 + a 4 U 4= > 

5 1 w 1 + b 2 u 2 -f b s u s + b 4 u 4 = 0. 

VI. If V t .M f = and ^6^=0 are the plane equations of two 
points not coincident, then ^tf^-f- \ ^b i u i = is the plane equation of 
any point on the line connecting the first two points. As \ takes all 
values, all points of a range can be thus obtained. 

VII. If 2)<W= 0, V^.= 0, and ^c { u { = are the plane equations 
of three points not in the same plane, then ^a i u i -} r \^b l u i +fj^c i u i = 
is the plane equation of any point on the plane determined by the first 
three points. As \ and p take all values, all points on the plane can 
be found. 



POINT AND PLANE COORDINATES 199 

The proofs of the last two theorems follow closely from theorems 
I and II of 83. 

The theorems of this section are plainly dualistic to the theorems 
of the previous section. We exhibit in parallel columns the funda 
mental dualistic objects: 

Point Plane 

Points in a plane. Planes through a point. 

Points in two planes. Planes through two points. 

A straight line. A straight line. 

Points of a range. Planes of a pencil. 

Planes of a bundle. Points of a plane. 

EXERCISES 

1. Write the equations, both in point and in plane coordinates, of the 
vertices, the faces, and the edges of the coordinate tetrahedron. 

2 . If a line is denned by the two points (^ : y z : y 3 : ?/ 4 ) and (z l : # 2 : # 8 : # 4 ), 
show that its equations in plane coordinates are 



and if a line is denned by the two planes (v l :v 2 :v s : v 4 ) and (w l :w 2 :w & : 
show that its equations in point coordinates are 



= 0. 

3. Show that the condition that two lines denned by the planes 

( a i : a : 3 : 4> ( b i b * :b * : ^) and ( c i :c * :c & : c ^> ( d i d 2 d a - d *), respec 
tively, should intersect is 



= 0, 



and write the similar condition for two lines, each denned by two 
points. 

4. Two conjugate imaginary lines being denned as lines such that 
each contains the conjugate imaginary point of any point of the other, 
show that if two conjugate imaginary lines intersect, the point of inter 
section and the plane of the two lines are real. Hence show that 
conjugate imaginary lines cannot lie on an imaginary plane. 



200 THREE-DIMENSIONAL GEOMETRY 

5. Show that if a plane contains two pairs of conjugate imaginary 
points which are not on the same straight line the plane is real. 

6. Two conjugate imaginary planes being defined as planes such that 
each contains the conjugate imaginary point of any point of the other, 
show that the plane coordinates of the planes are conjugate imaginary 
quantities, and conversely. Prove that two conjugate imaginary planes 
intersect in a real straight line. 

85. One-dimensional extents of points. Consider the equations 



where t is an independent variable and () are functions which 
are continuous and possess derivatives of at least the first two 
orders. We shall also assume that the ratios of the four functions 
f t (t) are not independent of t. Then, to any value of t corresponds 
one or more points xjxjxjx^ and as t varies these points describe 
a one-dimensional extent of points, which, by definition, is a curve. 
It is evident that because of the factor p the form of the functions 
fi(f) may be varied without changing the curve, but there is no 
loss of generality if we assume a definite form for ft(f) and take 

P =i. 

Let y i be a point P obtained by putting t = t l m (1), and let Q 
be a point obtained by putting t = ^ + At. Then the coordinates 
of Q are y t .+ A/., and the points P and Q determine a straight line 
with the equations 



or o-z^+XA^., (2) 

where the ratios of Ay { and not the separate values of these quantities 
are essential. As At approaches zero the ratios A^: A# 2 : Ay 3 : Ay 4 
approach limiting ratiosc^: dy. 2 : dy 8 : d&=./i(O : J^(*i) : ./J(*i) : ./I(*i> 
and the line (2) approaches as a limit the line 

px t = y f + My,=f,(td + VJft). (3) 

which is called the tangent line to the curve. At every point of the 
curve at which the four derivatives ./J(t) do not vanish, there is a 
definite tangent line. 



POINT AND PLANE COORDINATES 201 

The points y i and y i -f- dy { , which suffice to fix the tangent line, 
are often called consecutive points of the curve, but the exact 
meaning of this expression must be taken from the foregoing 
discussion. 

We shall now show that the tangent lines to a curve in the neigh 
borhood of a fixed point of the curve form a point extent of two dimen 
sions, unless in the neighborhood of the point in question the curve is a 
straight line. 

This follows in general from the fact that equations (3) involve 
two independent variables ^ and X. To examine the exceptional 
case we notice that at least two of the functions f. (t) cannot be 
identically zero if equations (1) do not represent a point. We 
shall also consider the neighborhood of a value ^ in which f t (f) 
are one-valued, and shall take/ 8 (T) and/ 4 (T) as the two functions 

which do not vanish identically. We may then place 3 ^ = T and 
replace equations (1) by the equivalent equations * 



= T 



(4) 



where ^(T) and f\(r) are one-valued in the neighborhood considered. 
The equations of the tangent line are then 



and the points on these lines form a two-dimensional extent unless 
^.(r 1 ) + X^(T 1 )=^.(T 1 +X). (i = l,2) (5) 

From this follows, by differentiating (5) with respect to X, 

JJOO-tfOi+X), (6) 

and by differentiating (5) with respect to T I? 

^(T 1 )+X^ (T 1 )=(T 1 +X), (7) 

and from (6) and (7) we have ^ (T,) = ; whence P t (T,)= e n r + c i2 . 



202 THKEE-DIMENSIONAL GEOMETKY 

Equations (4) then reduce to 



These are the equations of a straight line and the theorem is proved. 

Consider now three points, P, Q, A>, on the curve (1) with 
the coordinates y t , y t .+ Ay t ., and y.+ Ay.+ A(y + Ay ), the incre 
ments corresponding to the increment A; that is, 

&=/<(*i)> ft + A^/A + ^O. & + A&+A(y,+AyO =/<+ 2 A*). 
Then by the theorem of the mean, 

Ay =/,< 4- AO -/-ft) = CtfCO + O A, 
and by expansion into Maclaurin s series, 

2 A - 2 



The three points P, $, and 7? determine a plane whose coordi 
nates u { satisfy the three equations 

u &i+ u *+ Va+ M^ 4 = 0, 

w t Ay x + i/. 2 Ay 2 + M 8 Ay 8 + w 4 Ay 4 - 0, (8) 

w^^H- w, 2 A 2 y 2 + w 8 A 2 y 8 + w 4 A a y 4 = 0. 

As A approaches zero the three points P, Q, and-.fi approach 
coincidence, and the plane (8) approaches as a limit the plane 
whose coordinates satisfy the three equations 

^2/1+ W 2 #o +^3 + W 4#4= 

u l dy l + u 2 dy 2 -f w 3 ^ 3 + w 4 <% 4 - 0, (9) 

VVi+ W 2 + V 2 y 8 + W 4 = - 

This plane is called the osculating plane at the point P. It is 
evident that at any point P there is in general a definite osculating 
plane. The only exceptions occur when the point P is such that 
the solution of the equations (9) is indeterminate. Writing these 
equations with derivatives in place of differentials we have 



POINT AND PLANE COORDINATES 203 

and in order that the solution of these equations should be inde- 
terminant it is necessary and sufficient that t l should satisfy the 
equations formed by equating to zero all determinants of the third 
order formed from the matrix 

/,(,) 



If these equations have solutions they will be in general discrete 
values of ^ which give discrete points on the curve at which the 
osculating plane is indeterminate. To examine the character of a 
curve for which the osculating plane is everywhere indeterminate, 
it is convenient to take the equations of the curve in the form (4). 
Equations (10) then take the form 

O) + W 3 r+ W 4 = 0, 
+ u&(r) + u a = 0, (11) 



and these have an indeterminate solution when and only when 

Fj (r)=0, *! (T)=0. (12) 

If equations (9) are true for all values of T, the curve is a 
straight line, as has already been shown. 

Equations (10) determine u { as functions of the parameter t^ 
Therefore the osculating planes of a curve form in general a one- 
dimensional extent of planes. An exception can occur only when 
the ratios of u { determined by (10) are constant. To examine this 
case take again the special form (4) of the equations of the curve 
and consider equations (11). If the ratios u t determined by (11) 
are constant, it is first of all necessary that 

^ (T)=^ (T); 

whence ^2( T ) ^i^i( T ) + C -2 T + c s- 

Equations (4) then become 



204 THREE-DIMENSIONAL GEOMETRY 

and any point whose coordinates satisfy these equations lies in 
the plane ^-s.+ ^+^O. 

It is evident from the definition that this plane is the osculating 
plane at every point of the curve, and this can be verified from equa 
tions (11). We may accordingly make more precise the theorem 
already stated by saying that the osculating planes of a curve in the 
neighborhood of a fixed point of the curve form a one-dimensional extent 
of planes unless the curve is a plane curve in the neighborhood considered. 

If from equations (1) the parameter t is eliminated in two ways, 
there results two equations of the form 

/(^, x 2 , x 3 , * 4 )=0, 

9 0*v^ 2 8 a; 4)= - 

Conversely, any equations of form (13) may in general be replaced 
by equivalent equations of form (1). 

EXERCISES 

1. Show that in nonhomogeneous coordinates the equations of the 
tangent line and the osculating plane are, respectively, 

X-x Y-y Z -z 

dx dy dz 

X-x Y-y Z -z 



and 



dx dy dz 



= 0. 



2. Find the tangent line and osculating plane to the following curves : 

(1) The cubic, x = t s , y = t 2 , z = t. 

(2) The helix, x = a cos 0, y a sin 0, z = kO. 

(3) The conical helix, x = t cos t, y = t sin t, z = kt. 

3. Show that the osculating plane may be defined as the plane ap 
proached as a limit by a plane through the tangent line to the curve at 
a point P and through any other point P , as P approaches P. 

4. Show that the osculating plane may also be defined as the plane 
approached as a limit by a plane through a tangent line at P and parallel 
to a tangent line at P , the limit being taken as P approaches P. 

5. The principal normal to a curve is the line in the osculating plane 
perpendicular to the tangent at the point of contact ; the binormal is the 
line perpendicular to the tangent and to the principal normal. Find the 
equations of these normals. 



POINT AND PLANE COORDINATES 205 

86. Locus of an equation in point coordinates. Consider the 

equation /(*,,*,,*., O = . (1) 

where / is a homogeneous function of x^ x^ x^ and x^ which is 
continuous and has derivatives of at least the first two orders. 
Two of the ratios x l : x 2 : x s : x 4 can be assumed arbitrarily, and the 
third determined from the equation. The equation therefore defines 
a two-dimensional extent of points which by definition is called a 
surface. 

If / is an algebraic polynomial of degree n, the surface is called 
a surface of the nth order. Any straight line meets a surface of the 
nth order in n points or lies entirely on the surface. To prove this 
notice that a straight line is represented by equations of the form 

/w^ft+M-i 

where y { and z { are fixed points, and that these values of x. substi 
tuted in (1) give an equation of the nth order in X unless (1) is 
satisfied identically. 

A tangent line to a surface is defined as the limit line approached 
by the secant through two points of the surface as the two points 
approach coincidence. Let y { be the coordinates of a point P on 
the surface and y.+ A# t . those of a neighboring point Q also on the 
surface. The points P and Q determine a secant line, the equations 
of which are px t =y t +\(y ( + A*,,), 

which can also be written 

*>*,-= &+M&, (2) 

where the ratios of A?/ t . and not their individual values are essential. 
Now let the point Q approach the point P, moving on the surface, 
so that the ratios A^ : A?/ 2 : Ai/ 3 : A?/ 4 approach definite limiting ratios 
ty\ : dy 2 dy z : dy^ Then the line (2) approaches the limiting line 



which is a tangent line to the surface at the point P. 

If the four derivatives ~-i > do not all vanish, the 

fy ty* fy, fy 
ratios dy 1 : dy z : dy z : dy^ are bound only by the condition 

+dyd+ dyO. (4) 



206 THREE-DIMENSIONAL GEOMETRY 

By Euler s theorem for homogeneous functions we have, since 
y { satisfies equation (1), 



By virtue of (4) and (5) any point x i of (3) satisfies the equation 



This is the equation of a plane, and its coefficients depend only 
upon the coordinates of P and not on the ratios dy^ : dy z : dy^ : dy^. 

Hence all points on all tangent lines to the surface satisfy the 
equation (6). Equation (6), however, becomes illusive, and the dis 
cussion which led to it is impossible when P is such a point that 



Points which satisfy these equations are called singular points, 
and other points are called regular points. We have, then, the 
following theorem : 

All tangent lines to a surface at a regular point lie in a plane 
called the tangent plane, the equation of which is (6). 

In the equation (6) the point y { is called the point of tangency. 
Conversely, any line drawn in the tangent plane through the point 
of tangency is a tangent line. To prove this take z., any point 
in the plane (6). Then 

a/ df 3f cf 

*1 F^ ^F^ + *3 ^ + *4 /- = > 
fyl ^ 2 fy, ^4 

and the equations of the line through y { and z. are 

px i y i + X2;. 

But a point Q on the surface may be made to approach P in 
such a* way that dy^ : dy^ : dy & : dy^ = z l : z 2 : z 3 : z f since the only 
restriction on dy { is given by (4), which is satisfied by z.. Hence 
the line determined by y i and z. has equations of the form (3) and 
is therefore a tangent line, and the theorem is proved. 



POINT AND PLANE COORDINATES 



207 



The plane coordinates of the tangent plane to the surface (1) 
are, from (6), 

W,-f- ( = 1.2, 3, 4) (7) 

The coordinates y { can be eliminated between these equations, 

and the equation 

/O/i y<? y& # 4 ) (8) 

found by substituting y. for #. in (1). There are three possible 
results : 

1. There may be a single equation of the form 

This is the general case, in which the equations (7) can be 
solved and the results substituted in (8). 
The condition for this is that the Jacobian 



shall not vanish. In this case the tangent planes to (1) form a 
two-dimensional extent and their coordinates satisfy (9). 

If < (MJ, u 2 , u^ u^) is an algebraic polynomial of the rath degree, 
the surface (1) is said to be of the mih class. Through any straight 
line m planes can be passed, tangent to a surface of the mth class. To 
prove this notice that a plane through any straight line has the 

coordinates 

pu { = v { + \w., 

where v t . and w i are fixed coordinates. These values of u { substi 
tuted in (9) give an equation of the mih degree in \. This proves 
the theorem. 

For example, consider the surface 



208 THREE-DIMENSIONAL GEOMETRY 

The coordinates of its tangent plane are 



and these values substituted in 



a, yl = 



give 

^ 2 s 

The order and class of this surface are both 2, but the class of 
a surface is not in general equal to its order. 
2. There may be two equations of the form 



ifrfu^ u 2 , u 3 , w 4 )=0. 

In this case the tangent planes to (1) form a one-dimensional 
extent. The surface is called a developable surface. 
For example, consider the surface 

x* + x%- x%+ 2 x 3 x A - x*= 0. 
The coordinates of a tangent plane at y. are 



The elimination of y { from these equations and the equation 

yt+yt-yt+2y,yt-yt=v> 

gives the two equations u s -\- u = 0, 

<+<-<=0. 

3. There may be three equations of the form 



These equations can be solved for w v Hence in this case the 
tangent planes form a discrete system. 
For example, consider the surface 



POINT AND PLANE COORDINATES 209 

The tangent planes have the coordinates 



These lead to the equations 






= 0. 



The tangent planes are the two planes x 4 = and x^+x z +x n ~ 0. 
In fact the surface consists of these two planes. 

EXERCISES 

1. Show that the section of a surface made by a tangent plane is a 
curve which has a singular point at the point of contact of the plane. 

2. Show that the section of a surface of the nth order made hy any 
plane is a curve of the nth order. 

3. Show that any tangent plane to a surface of second order inter 
sects the surface in two straight lines, and in particular that the tangent 
plane to a sphere intersects the sphere in two minimum lines. 

4. Show that through the point of contact of a surface and a tan 
gent plane there go in general two lines lying in the plane and having 
three coincident points in common with the surface. 

5. Show that the equation f(x l9 x 2 , x a ) = 0, where the function / is 
homogeneous in x v x z , x s and the coordinate x 4 is missing, represents 
a cone, by showing that it is the locus of lines through the point 
0:0:0:1. 

6. Show that the tangent plane to a cone contains the element of 
the cone through the point of contact. 

7. From Ex. 5 show that in nonhomogeneous Cartesian coordinates 
the equation /(.x, y, z) = 0, where / is homogeneous, represents a cone 
with its vertex at the origin and that f(x, y) = represents a cylinder 
with its elements parallel to OZ. 

8. Show that through a singular point of a surface there goes in 
general a cone of lines each of which has three coincident points in 
common with the surface. 



210 



THREE-DIMENSIONAL GEOMETRY 



9. Find the equation or equations satisfied by the coordinates of 
the tangent planes of each of the following surfaces : 

(1) 2 ax,x 2 + l>x} + cxl = 0, 

(2) 2 cwc^a + ^ 2 2 + ex* = 0, 

(3) 2 axjXt + for? + c.rf = 0. 

10. Show that the tangent planes of a cone or a cylinder form a 
one-dimensional extent. 

11. If the equation of a surface is written in the nonhomogeneous 
form z =/(, y) t show that its tangent planes form a two-dimensional 

* 2 2 

extent unless rt s 2 0, where r= 



> $ = T^TT 
ex* dxdy 



t = 



12. Show that two simultaneous equations <^> 1 (^ 1 , x 2 , x 3 , a; 4 )= and 
^> 2 (x 1 , x 2 , # 3 , se 4 ) = define a curve, and that if the tangent planes to 
the curve are defined as the planes through the tangent lines to the 
curve, they form a two-dimensional extent given by the equations 

pU; = ~^ -f- XTT^ together with the equations of the curve. 
ox i ox i 

87. One-dimensional extents of planes. Consider the equations 

Wl =/!(0. 

(1) 




f. =/,(<) 

where u i are plane coordinates, ^ an independent variable, and 

fi(t) functions of t which are con 

tinuous and possess derivatives of 

at least the first two orders. We 

shall also assume that the ratios of 

the four f unctions /.(i) are not in 

dependent of t. The equations then 

define a one-dimensional extent of 

planes. Let v be the coordinates 

of a plane p (Fig. 53) obtained by 

placing t = 1 1 in (1) and let v t + Av t - 

be the coordinates of a plane q 

found by placing t = t l + A. Then p and q determine a straight 

line m, the equations of which are 




FIG. 53 



or 



pu { = v t + 
o-Ui v t + 



POINT AND PLANE COORDINATES 211 

As A approaches zero the line m approaches a limiting line /, 
of which the equations are 

M = v< + \d Vi =/. (g + \f( (g. (2) 

This line is called a characteristic of the extent denned by (1). 
It is evident that in any plane of the extent for which the four deriv 
atives f ^t) do not vanish there is a definite characteristic. 

We shall now prove the proposition 

The characteristics form in general a surface to which each plane of 
the defining plane extent is tangent along the entire characteristic in 
that plane. 

To prove this we notice that any point x i which lies in a char 
acteristic satisfies the two equations 

= o, 



,// (0 + *JXf) + * 3 



and that in general t may be eliminated from these equations with 
a result of the form , , \ A 

tC***?****) 8 *^ (4) 

This proves that any point on any characteristic lies on the sur 
face with the equation (4). 

By virtue of the manner in which (4) was derived we may write 



where t is to be determined as a function of x i from the second of 
equations (3). Therefore 



This shows that the tangent plane of (4) is the plane u. of the 
extent (1) and that the same tangent plane is found for all points 
for which t has the same value ; that is, for all points on the same 
characteristic. The proposition is then proved. 

Consider now three planes, v t ., v { -|-Av t ., v { + Av t . +A(v. +Av.). 
They determine a point P the coordinates of which satisfy the 
three equations ^ + Vi + x^x^= 0, 



z A v 2 + x s kv s + x^ A v 4 =0, (5) 

2 v 2 + a 8 A 2 r a + a- 4 A*v 4 = 0, 



212 THREE-DIMENSIONAL GEOMETRY 

and as A approaches zero the point P approaches as a limit a 
point L the coordinates of which satisfy the equations 



x^dv^ + x 2 dv 2 + x 3 dv 3 + x 4 dv 4 = 0, (6) 

x^\+ xd\ + x 3 d\+x 4 d\ = 0, 
or, what is the same thing, the equations 

3/ 3 (0 H- V 4 (0 = 0, 
/ 3 (0 + *4/J(0 = 0, (7) 

/3 (0 + xj*l (0 - o. 

The point L we shall call the limit point in the plane v i and shall 
prove the following proposition : 

The locus of the limit points is in general a curve, called the 
cuspidal edge, to which the characteristics are tangent. 

The first part of the proposition follows from the fact that equa 
tions (6) can in general be solved for x l as functions of t. 

To prove the second part of the proposition note that by differ 
entiating the first two equations of (7) on the hypothesis that 

x i> X 

three equations (7), we have 



2-> x s-> X 4 an( ^ * varv > an d reducing the results by aid of the 



Now from (3), 86, the tangent line to the cuspidal edge at a 
point (a: 1 , # 2 , # 8 , a? 4 ) given by a value t has the equations 



and from (7) and (8) any values of the coordinates X^ which satisfy 
these equations satisfy also 



that is, the point X { lies on the characteristic (3). 

To complete the general discussion we shall now prove the 
proposition 

The osculating planes of the cuspidal edge are the planes of the 
defining plane extent. 

By differentiating the first of equations (7) and reducing by 
the aid of the second equation, we have ^\dx i f i (^t^)=0. Therefore 



POINT AND PLANE COORDINATES 213 

by selecting the proper equations from (3) and (8) and replacing 
/i(0 by fl p we have the equations 



But from (9), 85, these equations define v i as the osculating 
plane of the cuspidal edge. This proves the proposition. 

In the foregoing discussion we have considered what happens in 
general. To examine the exceptional cases we may, as in 85, 
write the equations (1) in the form 



pu = 
H 2 



The equations (3) for the characteristics now become 
a^iCO + * a *iCO + 3 + ar 4 = 0, 
^ 1 (r) + rr 2 ^(T)4-^ 3 =0, 

and the equations (7) for the limit points become 

*i^i( T ) + ^O) + ay + ^ 4 = 0, 
^(T) + ^ (T) + ^ =0, (11) 

=0. 



The second of the equations (10) can be solved for T unless 



whence ^(r) = ^T + <? 8 , 

and ^(r)=0, 

In this case equations (10) become 



so that all characteristics are the same straight line. At the same time 
equations (9) become pu =cr + c , 



= T, 



which are of the type (2), 84, and represent a pencil of planes 
determined by the two planes (c g : c 4 : : 1) and (^ : c o : 1 : 0). The 



214 THREE-DIMENSIONAL GEOMETRY 

axis of the pencil is the straight line (12) with which the charac 
teristics coincide. 

Turning now to equations (11) we see that the last one deter 
mines x l : x 2 and the others determine x z and x^ unless F" (T) = 
and F" (T) 0. This is the same exceptional case just considered. 
The equations for the limit points become equations (12), so that 
the limit point in each plane is indeterminate but lies on the axis 
of the pencil of planes. 

Another exceptional case appears here also when the solutions 
of (11) do not involve r. This happens when 

*?<>>* V?<*>; 

whence ^ 2 ( T ) = c i-^i( T ) + c z r + *V 

Equations (11) then have the solution 

vvv**=v- 1: v<v 

At the same time equations (9) are 



PU 8 = T, 

pu^ = 1. 

All planes which satisfy these equations pass through the point (13). 

The surface of the characteristics is in this case a cone, since it 
is made up of lines through a common point. The cuspidal edge 
reduces to the vertex of the cone. 

In 86 we have shown that the tangent planes to a surface 
may, under certain conditions, form a one-dimensional extent of 
planes, and have called such surfaces developable surfaces. We may 
now state the following theorem, which is in a sense the converse 
of the above : 

Any one-dimensional extent of planes is composed of planes which 
are tangent to a developable surface, where, in the neighborhood of 
each point, the surface may be one of the following three kinds : 

1. It may be composed of tangent lines to a space curve. 

2. It may be a cone. (If the vertex is at infinity, the cone is a 

cylinder.) 

3. It may degenerate into the axis of a pencil of planes. 



POINT AND PLANE COORDINATES 215 

In the above theorem the nature of the surface has been de 
scribed only for each portion of it, since the foregoing discussion 
is based on the nature of the functions f { () in the neighborhood 
of a value of , which fixes a definite plane, a definite character 
istic, and a definite point on the cuspidal edge. In the simplest 
case the developable surface will have throughout one of the 
forms given above. Next in simplicity would be the case in which 
the surface is composed of two or more surfaces, each of which is 
one of the above kinds. It is of course possible to define surfaces 
which have different natures in different portions, but the char 
acter of each portion must be as above if the functions/^. () satisfy 
the conditions given. 

The planes of the extent are said in each case to envelop the 
developable surface. 

88. Locus of an equation in plane coordinates. Consider an 
equation u = 



where / is a homogeneous function of the plane coordinates u f . We 
shall consider only functions which are continuous and have deriva 
tives of at least the first two orders. Two of the ratios u^ : u 2 : u s : u^ 
can be assumed arbitrarily, and the third determined from the equa 
tion. Hence the equation represents an extent of two dimensions. 

If f is a polynomial of the nth degree, then n planes belonging to 
the extent (1) pass through any general line in space. The proof 
is as in 86. In this case the extent is 
said to be of the nib. class. 

We shall not restrict ourselves, how 
ever, to polynomials in the following dis 
cussion, but shall proceed to find some of 
the general properties of the extent (1). 

Let Vf be the coordinates of a plane p 
(Fig. 54) of the configuration defined by 
(1), and v i 4- Av t . those of another plane #, 

also of the configuration. The two planes p and q determine a 
line m whose equations in plane coordinates (theorem I, 84) are 

Pt9 < + X( 

or, otherwise written, (TU { v. + /-tAv,., 
where the ratios only of Av. are essential. 




216 THREE-DIMENSIONAL GEOMETRY 

Now let q approach coincidence with p in such a way that the 
ratios Av 1 : Ai> 2 : Av g : A# 4 approach limiting ratios dv^ : dv 2 : dv s : dv^. 
The line m approaches a limiting line L whose equations in plane 
coordinates are ffu . = v , + ^ v .. 

The differentials dv are bound only by the condition 



so that the planes with coordinates dv l : dv 2 : dv 3 : dv^ form a linear 
one-dimensional extent which by theorem II, 84, consists of all 
planes through the point JP, whose coordinates are 



This point lies in the plane v f since, by Euler s theorem for 
homogeneous functions, 



which is the condition (1), 84, for united position. 

A line L is the intersection of any one of the planes dv^i dv 2 : dv s : dv 4 
with the plane v 1 : v 2 : v s : t> 4 . Hence the lines L form a pencil of 
lines through P. 

The point P is not determined by equations (3) if 

Jf.O, M = 0, ff-A f = 0. (5) 

Wj <7V 2 <?V g <7V 4 

A plane for which these conditions is met is called a singular 
plane of the extent (1). Other planes are called regular planes. 
We sum up our results in the following theorem : 

In any regular plane p of the extent (1) there lies a definite point P 
whose coordinates are given by (3) and which has the property that 
any line of the pencil with the vertex P and in the plane p is the limit 
of the intersection of p and a neighboring plane. 

The point P may be called the limit point in the plane p. 
The elimination of v { from equations (3) and equation (1), written 
in v f , will give the locus of the points P. There are three cases : 
I. The elimination may give one and only one equation of the 

form *(*,** *.)=<). (6) 



POINT AND PLANE COOKDINATES 217 

The locus of p is then a surface. If the extent (1) is of the nth 
class, the surface (6) is also called a surface of the nth class. 
II. There may be two equations of the form 



The locus of P is then a curve. 

III. There may be three equations connecting x^ # 2 , x^ x^. The 
points P are then discrete points. 

We shall now show that the planes of (1) are tangent to the 
locus of P in such a manner that P is the point of tangency of 
the plane p, in which it lies. 

To prove this write equation (4) in the form 

A+Vi+Vi+ A-. 

and differentiate. We have 

^v,dx. +2^ t .cZv t .= 0, 
which, by aid of (2) and (3), is 



Consider now in order the previous cases. 

I. If x. satisfy a single equation (6), we have 



By comparison of (8) and (9) we have pv f = ^ which shows 

that v i are the coordinates of the tangent to < = at the point x t . 
II. If x i satisfy the two equations (7), we have 



A comparison with (8) gives pv . = + X ^ which shows that 

Vf passes through the line of intersection of the tangent planes to 
</\= and ( 2 = and hence is tangent to the curve denned by the 
two surfaces. 

III. If the points x. are discrete points, we may say that each 
plane of the extent is tangent to the point, through which it passes, 



218 THREE-DIMENSIONAL GEOMETRY 

thus extending the use of the word " tangent " in a manner which 
will be useful later. Summing up, we say : 

A two-dimensional extent of planes consists of planes which are 
tangent either to a surface or to a curve or to a point. 

The theorem has reference, of course, only to the neighborhood 
of a plane of the extent. The entire extent may have the same 
nature throughout or different natures in different portions. 

89. Change of coordinates. A tetrahedron of reference and a set 
of coordinates x t having been chosen, consider any four planes not 
meeting in a point the equations of which are 



the coefficients being subject to the single condition that their deter 
minant | a ik | shall not vanish. We assert that if we place 

px f = a^x, 4 a, 2 x 2 + a is x s + a^ , (2) 

then x[ are the coordinates of the point x i referred to the tetrahedron 
formed by the four planes (1). The proof runs along the same 
lines as that of the corresponding theorem in the plane ( 29) and 
will accordingly not be given. 

It is also easy to show that by the same change of the tetrahedron 
of reference, the coordinates u { become wj, where 

pu . = a liUl 4 a,.u, 4- a si u, + a 4i u, . (3) 

The change from one set of Cartesian coordinates to another is 
effected by means of formulas which are special cases of (2). If 
(x:y:z:t) are rectangular Cartesian coordinates and 

a^x + 1$ 4- c t z -h ef = 0, 

a 2 x + b z y + c z z + et = 0, (4) 

V + ty + V + V 830 

are any three iionparallel planes, and we place 

px = \(ajc + \y -f cz 4- e), 

py < = Ic^ajc 4- 1$ + c 2 2 + 2 0, 5 

pz = k s (a 3 x s 4- \y, 4- c s 8 4 ef), 

pt =t, 



POINT AND PLANE COORDINATES 219 

the quantities x\ y , z , t are proportional to the perpendiculars on 
the three planes, and it is possible to adjust the factors k i so that 
x ly iz i t may be exactly the Cartesian coordinates referred to the 
planes (4) as coordinate planes, the coordinates being rectangular 
or oblique according to the relative position of the planes (4). 

The equations (5) represent a change from a rectangular set of 
coordinates to another set which may or may not be rectangular, 
and conversely. A change from an oblique system to another is 
represented by formulas of the same type, since the change may 
be brought about as the result of two transformations of this type. 

EXERCISES 

1. Find the characteristics, characteristic surface, and cuspidal edge 
of each of the following extent of planes : 

(1) ^ = 1, pt( 2 = 3 #, P u s = 3 t\ pu^ = t*. 

(2) pWj= a/c sin t, pu 2 = ak cos t, pn z = a 2 , pi 4 = a*kt. 

(3) pu l= l - t*, P u 2 = 2 t, P u t =- 

(4) pu l =2 t, P u 2 = t 2 - 1, P w 8 

2. If a minimum developable is defined as a one-dimensional extent 
of minimum planes, show that the characteristics are minimum lines and 
the cuspidal edge is a minimum curve unless the developable is a cone. 

3. Show that the necessary and sufficient condition that the surface 
z=f(x, y) should be a minimum developable is that p 2 -f (f +1 0, 

where p = ^, q = TT- . (Compare Ex. 11, 86.) 

4. Prove that planes which are tangent at the same time to two 
given surfaces, two given curves, or a given surface and a given curve 
define developable surfaces. 

5. Find the envelope of each of the following one-dimensional extent 

of planes : 

(1) 2 ul + 3 ul + 4 ul - 24 y l = 0. 

(2) 3w 1 2 ? 8 -w 4 8 =0. 

(3) *+- ;=<>. 

(4) Wl 2 + nl + 2 ul - 2 Ul n 2 -f- 2 Wl w a - 2 2 w 8 - it; = 0. 

6. Show that the minimum planes form a two-dimensional extent 
and find its equation. 

7. Show that px t =f t (t) -\-sfi (t) (I = 1, 2, 3, 4) defines a developable 
surface and, conversely, that any developable surface which is not a 
cone or the axis of a pencil of planes may be expressed in this way. 



CHAPTER XIII 

SURFACES OF SECOND ORDER AND OF SECOND CLASS 
90. Surfaces of second order. Consider the equation 



which defines a surface of second order ( 86). The Jacobian of 
86 becomes, except for a factor 2, the determinant 



a u % 



called the discriminant of the equation. We may make the follow 
ing preliminary classification : 

I. A = 0. The surface has a doubly infinite set of tangent planes. 
The plane equation of the surface may be found by eliminating u { 
from the equations 



(2) 



and equation (1). But a combination of (2) and (1) gives readily 



and the elimination of o^. from this equation and the set (2) gives 



i a a u 

22 23 24 1 



u 4 



= 0. 



(3) 



220 



SUKFACES OF SECOND ORDER AND SECOND CLASS 221 

This is an equation of the second degree in u t . Hence a sur 
face of the second order for which the discriminant is not zero is also 
a surface of the second class ( 88). 

It is not difficult to show that the discriminant of (3) is not 
equal to zero. 

II. A= 0. The tangent planes either form a one-dimensional 
extent of planes or consist of discrete planes. These cases will be 
examined later. 

91. Singular points. By 86 singular points on the surface (1), 
90, are given by the equations 



There are four cases : 

I. A = 0. Equations (1) have no solution, and the surface has 
no singular points. This is the general case. 

II. A= 0, but not all its first minors are zero. The surface has 
one and only one singular point. Let y. be the coordinates of the 
singular point and z- t the coordinates of any other point in space, 
and consider the straight line 

px .= yi +\ Zi . (2) 

To find the points in which the line (2) meets the surface sub 
stitute in equation (1), 90. Since the coordinates y. satisfy the 
equation of the surface and also the equations (1), the result is 

V2,W t =0- (3) 

This shows that any line through a singular point meets the sur 
face only at that point (X = 0), and there with a doubly counted 
point of intersection. An exception occurs when 2. is taken on 
the surface. Then equation (3) is identically satisfied, and the 
line yz lies entirely on the surface. Hence the surface is a cone 
ivith the singular point as the vertex. There is no plane equa 
tion of the surface. In fact the tangent planes form a singly 
infinite extent of planes, and their coordinates are subject to two 
conditions. 



222 THREE-DIMENSIONAL GEOMETRY 

III. A = 0, all its first minors are zero, but not all its second minors 
are zero. Equations (1) contain two and only two independent equa 
tions and hence the surface has a line of singular points. If this 
line is taken as the line x^= 0, x 2 = in the coordinate system, equa 
tions (1) show that we shall have a is = a u = a zs = a^ = a ss = a s4 = a =0, 
and the equation of the surface becomes 11 # 1 2 + 2 12 ^ 2 + 22 ^ 2 2 = 0. 
At least two of the coefficients in the last equation cannot vanish, 
since the surface has only the line 2^=0 and x^= of singular 
points. Therefore the left-hand member of the equation of the sur 
face factors into two linear factors. Hence the surface consists of 
two distinct planes intersecting in the line of singular points. 

IV. A= 0, all its first and second minors are zero, but not all 
the third minors are zero. Equations (1) contain one and only one 
independent equation, and hence the surface has a plane of sin 
gular points. If this plane is taken as x 1 = 0, the equation of the 
surface becomes x* = 0. Hence the surface consists of the plane of 
singular points doubly reckoned. 

92. Poles and polars. The polar plane of a point y i (the pole) 
with respect to a surface of the second order whose equation is 
(1), 90, is defined as the plane whose coordinates are 

a, 2 y 2 + a^ + a^y, . (1) 




The following theorems are obvious or may be proved as are 
the similar theorems of 34 : 

/. If the pole is on the surface, the polar plane is a tangent plane, 
the pole being the point of contact. 

II. To every point not a singular point of the surface corresponds 
a unique polar plane. 

III. To every plane corresponds a unique pole tvhen and only when 
the discriminant of the surface does not vanish. 

IV. A polar plane contains its pole when and only when the pole is 
on the surface. 

V. All polar planes pass through all the singular points of the 
surface when such exist. 

VI. If a point P lies on the polar plane of a point Q, then Q lies 
on the polar plane of P. 

* VII. All tangent planes through a point P touch the surface in a 
curve which lies in the polar plane of P. 



SURFACES OF SECOND ORDER AND SECOND CLASS 223 

VIII. For a surface of second order without singular points it is 
possible in an infinite number of ways to construct a tetrahedron in 
ivhich each face is the polar plane of the opposite vertex. 

These are self-polar tetrahedrons. 

IX. If any straight line m is passed through a point P, and R and 
S are the points in which m intersects a surface of second order and 
Q is the point of intersection of m and the polar plane of P, then P 
and Q are harmonic conjugates with respect to R and S. 

In addition to these theorems we will state and prove the 
following, which have no counterparts in 34: 

X. The polar planes of points on a range form a pencil of planes the 
axis of which is called the conjugate polar line of the base of the range. 
Reciprocally the polar planes of points on the axis of this pencil form 
another pencil the axis of which is the base of the original range. 

Consider any range two of whose points are P and Q (Fig. 55). 
Let the polar planes of P and Q intersect in LK, and let A be any 
point of LK. The polar plane of A must contain both P and Q 
(theorem VI) and hence the entire line PQ. Now let R be any 
point on PQ. Its polar plane must 
contain A (theorem VI). But A is 
any point of LK. Therefore the polar 
plane of R contains LK. This proves 
the theorem. It is to be noted that the 
opposite edges of a self-polar tetra 
hedron are conjugate polar lines. 

XL If two conjugate polar lines in 
tersect, each, is tangent to the surface 
at their point of intersection. K \Q 

Let two conjugate polar lines, PQ FIG 55 

and LK, intersect at R. Since R 

lies in each of the lines PQ and LK its polar plane must contain 
each of these lines by the definition of conjugate polar lines. Hence 
the polar plane of R contains R and is therefore (theorems IV 
and I) the tangent plane at R. The two lines LK and PQ lying 
in the tangent plane and passing through R are tangent to the 
surface at R. 




224 THREE-DIMENSIONAL GEOMETRY 

EXERCISES 

1. Show that any chord drawn through a fixed point P 9 intersecting 
at infinity the polar plane of P with respect to a quadric, is bisected by P. 
Hence show that if a quadric is not tangent to the plane at infinity there 
is a point such that all chords through it are bisected by it. This is 
the center of the quadric. 

2. Show that the locus of the middle points of a system of parallel 
chords is a plane which is the polar plane of the point in which the 
parallel chords meet the plane at infinity. This is a diametral plane 
conjugate to the direction of the parallel chords. Show that a diametral 
plane passes through the center of the quadric, if there is one, and 
through the point of contact with the plane at infinity if the surface 
is tangent to the plane at infinity. 

3. Prove that all points on a straight line which passes through the 
vertex of a cone have the same polar plane ; namely, the diametral plane 
conjugate to the direction of the line. 

4. Show that if a plane conjugate to a given direction is parallel to 
a second given line, the plane conjugate to the latter line is parallel to 
the first. Three diametral planes are said to be conjugate when each 
is conjugate to the intersection of the other two. Show that the inter 
sections of three conjugate diametral planes with the plane at infinity 
form a triangle which is self polar with respect to the curve of inter 
section of the quadric and the plane at infinity. Discuss the existence 
and number of such conjugate planes in the two cases of central quad- 
rics and quadrics tangent to the plane at infinity. 

5. Show that if a line is tangent to a quadric surface its conjugate 
polar is also tangent to the surface at the same point, and that the two 
conjugate polars are harmonic conjugates with respect to the two lines in 
which the tangent plane at their point of intersection cuts the surface. 

6. Show that the conjugate polars of all lines in a pencil form a 
pencil. When do the two pencils coincide ? Show that the conjugate 
polars of all lines in a plane form a bundle of lines, and conversely. 

93. Classification of surfaces of second order. With the aid of the 

results of the last two sections it is now possible to obtain the 
simplest equations of the various types of surfaces of the second 
order which have already been arranged in classes in 91. 

I. The general surface. A = 0. The surface has no singular point 
(91) and there can be found self -polar tetrahedrons (92). Let 
one such tetrahedron be taken as the tetrahedron of reference in the 



[JRFACES OF SECOND ORDEK AND SECOND CLASS 225 

coordinate system. Then the equation of the surface must be such 
thai the polar of : : : 1 is x^= 0, that of : : 1 : is x s = 0, 
tha, of : 1 : : is x 2 = 0, and that of 1 : : : is x^ = 0. The 
equation is then 2 2 2 Q 2 

11 1 22 2 ~T~ 33 3 44 4 "j V X 



wh;re no one of the coefficients can be zer^^br, if it were, the 
surface would contain a singular point. ^B 

It is obvious that if the original tetralflj^n of reference were 

real and if the coefficients in the original equation of the surface 

were real, the new tetrahedron of reference and the new coefficients 

are also real. We may now replace x i in the last equation by a> i \x i 

^arid have three types according to the signs of the terms resulting. 

it 1. The imaginary type, xl -f xl -+- xl -t- T?= 0. (3) 

Th v is equation is satisfied by no real points. 

2.\The oval type, x? + x* + xl~ x?= 0. (4) 

No r l -eal straight line can meet this surface in more tl\ n two real 
points. If it did, it would lie entirely on the surface (/>), and 
hence tht^ point in which it met the plane x 4 = would be a real 
point of tl^ie surface. But the plane x^= meets the surface in the 
curve xl+ A *;-\-x= 0, which has no real point. Hence, as was said, 
no real stra ight line can meet the surface in more than two real 
points. The surface, however, contains imaginary straight lines as 

will be seen li iter. 



3. The saddle type, xl + xl - xl - xl = 0. (4) 

Through every , point of this surface go two real straight lines 
which lie entirel} - on the surface. This follows from the fact that 
whatever be the v alues of \ and /-i, the two lines 

lie entirely on the surfae^e. Moreover, values of X and //, may be 
easily found so that one o% each of these straight lines may pass 
through any point of the surface. This matter will be discussed in 
detail in 96. 

As the three types of surfaces * here named are distinguished by 
properties which are essentially diti^erent in the domain of reality, 




226 THREE-DIMENSIONAL GEOMETRY 

the corresponding equations can evidently not be reduced to :ach 
other by any real change of coordinates. However, if no distinction 
is made between reals and imaginaries, all surfaces of the tiree 
types may be represented by the single equation 

xl + xl H- xl + xl = 0. (5) 

II. The cones. B^ but not all the first minors are zero. The 
surface has one shfljppoint ( 91) anc ^ ^ s a cone w ^ n ^ ne singular 
point as the vertex. Let the vertex be taken as A (0:0:0:1). 
Then in the equation of the surface a u = a^ = a^ = a^= 0. Take 
now as B (0 : : 1 : 0) any point not on the surface. Its polar plane 
contains A (theorem V, 92) but not B (theorem IV, 92). Take 
as C (0 : 1 : : 0) any poir in this plane but not on the surfac^. 
Such points exist unles? the polar plane of B lies entirely on t/?he 
surface, which is imp^jsible since B was taken as not on the surf * ace. 
The polar plane of C contains A and B and intersects the polar 
plane of B in a line through A. Take D (1 : : : 0) as anv} point 
on this Ijtfe. We have now fixed the tetrahedron of reference so 
that (? : : : 1 is a singular point, the polar plane of : Jf J : 1 : is 
x 9 -- 0, the polar plane of : 1 : : is x 2 = 0, and the r&olar plane 
of 1 : : : is x^= 0. Therefore the equation of thf/3 surface is 

a 11 x 1 + a^ a + a u x 9 , ^ ^ 

where no one of the three coefficients can vanish, sin/ce the surface 
has only one singular point. By a real transformatior^i of coordinates 
this equation reduces to two typks : 

1. The imaginary cone, x* -f xl + xl = 0. 

2. The real cone, x* + xl - xl = 0. 

III. Two intersecting planes. A= 0, all the fii/ st minors are zero, 
but not all the second minors are zero. This ff has been sufficiently 
discussed (91). There are obviously twcK> types in the domain 
of reals ; namely : 

1. Imaginary planes, ^ + # 2 2 =/ 0. 

2. Real planes, x? %lf 0- 

IV. Two coincident planes. A = O^f all the first and all the second 
minors are equal to zero. Evidently the equation in this case is 

reducible to the form jl^ n 

r^i ^i 







; 



SURFACES OF SECOND ORDER AND SECOND CLASS 227 

but the plane 2^= is not necessarily real. In fact the condition 
that all the second minors of A vanish is the condition that the 
left-hand member of equation (1), 90, should be a perfect square, 
as is easily verified by the student. 

94. Surfaces of second order in Cartesian coordinates. As we 
have seen ( 82), we obtain Cartesian coor^rates from general 
quadriplanar coordinates by taking one of y^i-Brdinate planes as 
the plane at infinity and giving special va^^o the constants k r 
This being done, the general equation of the second degree will 
be written 

a% 2 + bx 2 + cz*+ %fyz + Zgzx +2hxy + 2lxt+2 myt + 2nzt + dt*= 0, (1) 

which reduces to the usual nonhomogeneous form when t is placed 
equal to 1. 

For equation (1) the results of 90-93 remain unchanged 
except for a slight change of notation. We will refer to the equa 
tions of these sections by number and make the necessary change 
in notation without further remark. Assuming that A = we 
may find the pole of the plane at infinity, for example, by placing 
w t . in equations (1), 92, equal to the coordinates 0:0:0:1 of the 
plane at infinity. There result the equations 

ax + Tiy + gz + It = 0, 
lix + by +fz + mt = Q, 
gx +fy + cz + nt = 0, 
Ix 4- my + ftz + dt = /?, 

the solution of which is the coordinates of the pole required. This 
pole is therefore a finite point when the determinant 

a h 

D= h b f 
9 f c 
is not zero and is a point at infinity when D = 0. 

In the latter case, by theorems IV and I, 92, the surface is 
tangent to the plane at infinity. In the former case, if the pole 
of the plane at infinity is taken as 0:0:0:1, then l = m = n = 0, 
and consequently it appears that if xjyjz^.t^ is a point on the 
surface, x l : y l : z l : ^ is also on the surface. The point is 
therefore called the center of the surface, and the surface is called 



228 THREE-DIMENSIONAL GEOMETEY 

a central surface. Conversely, if a surface without singular points 
has a center (that is, if there exists a point which is the middle 
point of all chords through it), that point is the pole of the plane 
at infinity. This follows from theorem IX, 92, or may be shown 
by assuming the center as the origin of coordinates and reversing 
the argument jus^jaade. 

We have reac HH| following result : 

A surface of sec^fm der with the equatiomQ*) is a central surface 
or a noncentral surface according as the determinant D is not or is 
equal to zero. A noncentral surface is tangent to the plane at infinity. 

Holding now to the significance of the determinant A as given 
in 90 we may proceed to find the simplest forms of the equa 
tions of the surface in Cartesian coordinates. There will be this 
difference from the work of 93 that now the plane t = plays 
a unique role and must always remain as one of the coordinate 
planes. The other three coordinate planes, however, may be 
taken at pleasure, and we shall not at present restrict ourselves 
to rectangular coordinates. 

1. Central surfaces without singular points. As in 93, by refer 
ring the surface to a self-polar tetrahedron one of whose faces is 
the plane at infinity its equation becomes 



According to the signs of the coefficients this gives the following 
types in nonhomogeneous form: 

1. The oval type: 

x 2 if z 2 
(a) The imaginary ellipsoid, ~2+72+~2 == ~~l* 

222 

(6) The real ellipsoid, - + f- 2 + ^= 1. 

a" b 2 <f 



X 2 if" 2 2 

(c) The hyperboloid of two sheets, 2 ~ ^=1 

a o ct 

2. The saddle type: 

2 .2 2 

The hyperboloid of one sheet, ^+T^~~ ~^ = 1 

or I c 2 

II. Noncentral surfaces without singular points. Since the plane 
at infinity can no longer be a face of a self-polar tetrahedron, we 
cannot use the method of 93. We will take the point of tangency 



SURFACES OF SECOND ORDER AND SECOND CLASS 229 

in the plane at infinity as B (0:0:1:0). Then g =f= c = and 
n = 0. Take an arbitrary line through B. It meets the surface 
in one other point A, which we take as 0:0:0:1. We then take 
the tangent plane at A as 2 = 0. Then I = m d = 0, and the 
equation of the surface is 



z =Q. ,< 

The tangent plane at A meets the plane at infinity in a line 
( 2 =0, t = 0), which is the conjugate polar to the line AB (x = 0, 
y = 0). The points C (0 :1: : 0) and D (1: : : 0) may be taken 
as any two points on this line such that each lies in the polar 
plane of the other. Then h = 0, and the equation of the surface is 
reduced to 



712 = 0. 
According to the signs which occur we have two types : 

1. The oval type: 

x 2 if 
The elliptic paraboloid, -f- ^= nz. 

2. The saddle type: 

x 2 y 2 
The hyperbolic paraboloid, ^ = nz. 

The discussion of surfaces with singular points presents no features 
essentially different in Cartesian coordinates from those found in 
the general case. If the surface has one singular point, it is a cone 
if the singular point is not at infinity and is a cylinder if the sin 
gular point is at infinity. If the surface has a line of singular 
points, it consists of two intersecting or two parallel planes accord 
ing as the singular line lies in finite space or at infinity. If the 
surface has a plane of singular points, it consists of a plane doubly 
counted, which may be the plane at infinity. 

95. Surfaces of second order referred to rectangular axes. In the 
previous section no hypotheses were made as to the angles at which 
the coordinate planes intersected. For that reason the coordinate 
planes leading to the simple forms of the equations could be chosen 
in an infinite number of ways. We shall now ask whether, among 
these planes, there exist a set in which the planes x = 0, y = 0, 
and 2 = are mutually orthogonal. 



230 THREE-DIMENSIOHAL GEOMETRY 

Consider first the central surfaces without singular points for 
which I) = 0. The plane at infinity cuts this surface in the gen 
eral conic ^ + ^ + ^ 2 + 2 ^ + 2 ffgx + 2 h xy = o, (1) 



where x : y : z are homogeneous coordinates on the plane t = 0. 

When the equation of the surface is referred to a self-polar tetra 
hedron of which the plane at infinity is one face, the curve (1) is 
referred to a self -polar triangle. If the axes in space are orthogonal, 
the triangle must also be a self-polar triangle (theorem V, 81) 
to the circle at infinity 



" A 
ar=0. (2) 

Our problem, therefore, is to find on the plane at infinity a triangle 
which is self polar at the same time with respect to (1) and (2). 

By 43 this can be done when and only when the curves (1) 
and (2) intersect in four distinct points or are tangent in two 
distinct points or are coincident. 

In the first case there exists one and only one self-polar triangle 
common to (1) and (2), and therefore there exists only one set of 
mutually orthogonal planes passing through the center of the quad- 
ric and such that by use of them as coordinate planes the equa 
tion of the quadric becomes 

ax*+ bf + cz 2 + d=Q. (a = b = c = 0) 

These planes are the principal diametral planes of the quadric, 
and their intersections are the principal axes. 

In the second case there are an infinite number of planes through 
the origin, such that by use of them as coordinate planes the equa 
tion of the quadric becomes 

a(a?+y a )+c a +cJ = 0. (a = c = 0) 

Here the axis OZ is fixed, but the axes OX and Y are so far 
indeterminate that they may be any two lines perpendicular to OZ 
and to each other. The surface is a surface formed by revolving 
the conic ax*+ cz* + d= 0, y = about OZ. 

In the third case any set of mutually perpendicular planes through 
the origin, if taken as coordinate planes, reduce the equation of the 
quadric to the form 



and the quadric is a sphere. 



SURFACES OF SECOND OKDEK AND SECOND CLASS 231 

It is to be noticed that if the coefficients in equation (2) are 
real, one of the above cases necessarily occurs. For in this case 
the solutions of equations (1) and (2) consist of imaginary points 
which occur in pairs as complex imaginary points. 

If we consider the noncentral quadrics without singular points 
and use the notation of 94, we notice first that if the axes of 
coordinates are rectangular, the point B cannot be on the circle at 
infinity, since the line CD must be the polar of B with respect to 
the circle at infinity. The point B being fixed by the quadric sur 
face, the line CD is then fixed, and consequently the line AB, since 
AB is the conjugate polar of CD with respect to the quadric. The 
point A is then fixed and is called the vertex of the quadric. 

The points C and D must now be taken as conjugate, both with 
respect to the circle at infinity and with respect to the conic of inter 
section of the quadric and the plane at infinity. If the two straight 
lines into which this latter conic degenerates (cf. Ex. 1, 86) are 
neither of them tangent to the circle at infinity, the points C and 
D are uniquely fixed. If both of these lines are tangent to the cir 
cle at infinity, the point C may be taken at pleasure on (7Z>, and D 
is then fixed. 

In the first case there is one tangent plane and two other planes 
perpendicular to it and to each other, by the use of which the equa 
tion of the quadric is reduced to the form 

ax 2 + fy 2 = nz. (a = b) 

In the second case there are an infinite number of mutually 
orthogonal planes, one of which is a fixed tangent plane, by the 
use of which the equation of the quadric is reduced to the form 



and the quadric is a paraboloid of revolution. 

In all other cases, namely, when the point of tangency of the 
quadric with the plane at infinity is on the circle at infinity or 
when the section of the quadric with the plane at infinity consists 
of two straight lines, one and only one of which is tangent to the 
circle at infinity, the equation of the surface cannot be reduced to 
the above forms by the use of rectangular axes. 

If the coefficients of the terms of the second order in the equation 
of the quadric are real, the rectangular axes always exist. 



232 THREE-DIMENSIONAL GEOMETEY 

EXERCISES 

Examine the following surfaces for the existence of principal axes : 



2. 2 
3. 

4. 2 
5. 

6. x*+ 2 foy - 2/ 2 - 2 + 2z = 0. 

7. xz + iys + aj = 0. 

9. Examine the quadrics with singular points by the methods of 
this section. 

96. Rulings on surfaces of second order. We have seen (93) 
that the equation of any surface of tjie second order without 
singular points can be written as 

#1 + # 2 4- ^3 + X \ (1) 

if no distinction is made between reals and imaginaries or between 
the plane at infinity and any other plane. This equation can be 
written in either of the two forms 

_ 2l ^ = ^^, =X) (2) 

a"* 4 i 2 

*i + 



whence follows for any point on the surface 



From these equations the following theorems are easily proved : 

/. On a surface of second order without singular points lie two 
families of straight lines, one defined by equations (2) and the other 
by equations (3). 

For if X is given any constant value in (2) the equations 
represent a straight line every point of which satisfies equation (1). 
Similarly, ^ may be given a constant value in (3). The straight 
lines (2) and (3) are called generators. 



\ 



SURFACES OF SECOND ORDER AND SECOND CLASS 233 

//. Through each point of the surface goes one and only one line 
of each family. 

For any point x i of the surface determines X and p uniquely. 
///. Each line of one family intersects each line of the other family. 
For any pair of values of X and //. leads to the solution (4). 

IV. No two lines of the same family intersect. 
This is a corollary to theorem II. 

V. A tangent plane at any point of the surface intersects the sur 
face in the two generators through that point. 

For the two generators are tangents and hence lie in the tangent 
plane. But the intersection of the tangent plane with the surface 
is a curve of second order unless the plane lies entirely on the 
surface, which is impossible since the surface has no singular points. 
Hence the section consists of the two generators. 

VI. The surface contains no other straight lines than the generators. 

For if there were another line the tangent plane at any point 
of the line would contain it, which is impossible by theorem V. 

VII. Any plane through a generator intersects the surface also in 
a generator of the other family and is tangent to the surface at the 
point of intersection of the two generators. 

Consider a plane through a generator g. Its intersection with 
the surface is a curve of second order of which one part is known 
to be g. The remaining part must also be a straight line h, which 
is a generator by theorem VI. Since h and g are in the same plane 
they intersect and hence belong to different families by theorem IV. 
The tangent plane at the intersection of h and g contains these 
lines by theorem V and hence coincides with the original plane. 

VIII. If two pencils of planes with their axes generators of the 
same family are brought into a one-to-one correspondence so that two 
corresponding planes intersect in a generator of the other family, the 
relation is protective. 

Let the axes of the two pencils be taken as 2^=0, x 2 and 
x s = 0, x 4 = respectively. Since these lines lie on the surface, the 
equation of the surface has the form 



234 THREE-DIMENSIONAL GEOMETRY 

The equations of planes of the first pencil are 

Xl +\x 9 =Q 
and those of the second are 

X S + ^4 = 

If two such planes intersect on the surface, we have 



which proves the theorem. 

IX. The intersections of the corresponding planes of two protective 
pencils of planes with nonintersecting axes generate a surface of second 
order which contains the two axes of the pencils. 

Let the two pencils be 0^+ \% 2 = and 3 + /-t# 4 = 0, where the 

protective relation is expressed by X = - - 

7/* + S 

Then if a point is common to two corresponding planes, it 
satisfies the equation 

72:^+ axjc z - Bx t x 4 - fa 2 z 4 = 0, 

which is also satisfied by the axes of the pencils. 

X. (Dualistic to VIII.) Lines of one family of generators cut out 
projective ranges on any two lines of the other family. 

As in the proof of theorem VIII, let 2^== 0, x 2 be a generator 
of the surface and let # 3 = 0, x 4 = be another generator of the 
same family. The equation of the surface is then 



and the generators of the second family are 



A generator of this family meets ^=0, % 2 = in the point where 
2? 8 : x^ = c 4 + (? 2 \ : <? 3 CjX and meets x s = 0, x 4 = where x l : x 2 = X : 1. 
The relation is evidently projective. 

XI. (Dualistic to X.) The lines which connect corresponding points 
of two projective ranges with nonintersecting bases lie on a surface 
of second order. 



SURFACES OF SECOND ORDER AND SECOND CLASS 235 

Let one range be taken on ^=0, # 2 = and the other on x s = 0, 
x 4 =Q. Then the points of the two ranges are given on each base 
by the equations x z +\x^= and 2^+ /A# 2 = 0. Let the protective 

relation be expressed by X = - - 

7/* + S 

From these it is easy to compute that the coordinates of any point 
on the line connecting two corresponding points of the two ranges 
satisfy the equation 

t 0. 



EXERCISES 

1. Distinguish between the cases in which the generators are real or 
imaginary, assuming that the equation of the quadric is real. 

2. What are the generators of a sphere ? 

3. Distinguish between a central quadric and a noncentral one by 
showing that for the latter type the generators are parallel to a plane 
and for the former they are not. 

97. Surfaces of second class. Consider the equation 

]4^ t =0, (A k; =A ik ) (1) 

in plane coordinates. This is a special case of the equation dis 
cussed in 88. Equations (3), 88, which determine the limit 
points, become 

px i = A il u l + A i ,u. 2 -{-A i3 u 3 + A i ,u,, (i = l, 2, 3, 4) (2) 
and equations (5), 88, which define the singular planes, become 

4 1 M 1 + 4- 2 w a + 4 3 tt3 + 4 4 M4= - (* =1 2 3 4 ) 

If we now place 



A = 



A n 



we have to distinguish four cases. 

I. A^=0. Equations (2) have then a single solution for w t : u 2 :u s : u^ 
which, if substituted in (1), gives the equation of the surface en 
veloped by the extent of planes. This equation may be more con 
veniently obtained by replacing (1) by the equation 



236 



THREE-DIMENSIONAL GEOMETRY 



obtained from (1) by the help of (2). The elimination of u t then 
gives 



A Vi A 22 



A 



A \Z A 1Z A 33 ^34 X S 



= 0, 



(5) 



which is the equation of a surface of second order. 

Under the hypothesis A^ equations (3) have no solution, so 
that in this case no singular plane exists. ^ It is not difficult to 
show that the discriminant of equation (^) does not vanish. - 

We have, accordingly, the following result: A plane extent of 
second class with nonvanisJiing discriminant consists of planes envelop 
ing a surface of second order without singular points. 

This theorem may be otherwise expressed as follows : A surface 
of second class without singular planes is also a surface of second order 
without singular points. 

II. A = 0, but not all the first minors are zero. Equations (3) now 
have one and only one solution, so that the extent (1) has one and 
only one singular plane. Let it be taken as the plane 0:0:0:1. 
Then A u = A u = A M = A 4t =Oj and equation (1) takes the form 



A n u? -h A^ul + A^ul + 2 A^u^ + 2 A 1B UjU B + 2 A 2S u 2 u s = 0, (6) 
where the determinant 



11 



[ 12 4. 



^12 ^22 ^23 
4. ^,3 ^33 

does not vanish owing to the hypothesis that not all the first 
minors of the discriminant (4) vanish. 

The elimination of u { from equations (2) and equation (6) 
gives, then, 

, A U x i 

A /y 

I A 33 X 3 

x. 

which are the equations of a nondegenerate conic in the plane 
z 4 =0. 



SURFACES OF SECOND ORDER AND SECOND CLASS 237 

We have, accordingly, the result : A plane extent of second class 
with one singular plane consists of planes which are tangent to a non- 
degenerate conic lying in the singular plane. 

The equation of the plane extent may be considered the equa 
tion of this conic in plane coordinates. 

III. A= 0, all the first minors are zero, but not all the second 
minors are zero. Equations (3) now contain only two independent 
equations and hence the extent contains a pencil of singular planes. 
If this pencil is taken as u^ 0, u 2 = 0, the equation of the extent 

4X + 2 A l2 u^ + ^X =0, (7) 

where the determinant A U A M A^ does not vanish because of the 
hypothesis that not all the second minors of the discriminant (4) 
vanish. 

Equation (7) factors into two distinct linear factors and hence 
the plane extent consists of two bundles of planes. The elimina 
tion of u { between equations (2) and (7) gives 



which define the vertices of the two bundles. 

We have, accordingly, the result : A plane extent of second class 
with a pencil of singular planes consists of two bundles of planes, the 
singular pencil being the pencil common to the two bundles. 

IV. A= 0, all the first and second minors are zero, but not all 
the third minors are zero. Equations (3) contain only one inde 
pendent equation and hence the plane extent contains a bundle of 
singular planes. If this bundle is taken as u l = 0, the equation of 
the extent becomes 



where A n cannot be zero because of the hypothesis that not all 
third minors of (4) are zero. 

Hence we have the result : A plane extent of second class with a 
bundle of singular planes consists of that bundle doubly reckoned. 

It may be noticed that the elimination of u { between equations 
(2) and (8) gives the meaningless result x^: x z : x 8 : x = : : : 0. 



238 THKEE-DIMENSIONAL GEOMETRY 

98. Poles and polars. The relation between poles and polars may 
be established by means of plane coordinates as well as by point 
coordinates. We shall define the pole of a plane v t with respect to 
the extent (1), 97, as the point the coordinates of which are 

px t = A^-l- A { . 2 v 2 + A i3 v s + A^v 4 . (i = 1, 2, 3, 4) 

For the case in which A = the relation between pole and polar 
is the same as that defined in 92, as the student may easily prove. 
In the cases in which A = the polar relation is something new. 

The following theorems dualistic to those of 92 are obvious or 
easily proved : 

I. If a plane belongs to the extent its pole is the limit point in the 
plane. 

II. To any plane not a singular plane of the extent corresponds a 
unique pole. 

III. To any point corresponds a unique polar when and only when 
the plane extent has no singular plane. 

IV. A pole lies in its polar plane when and only ivhen the polar 
plane belongs to the extent. 

V. The pole of any plane lies in all singular planes when such exist. 

VI. If a plane p passes through the pole of a plane q, then q passes 
through the pole of p. 

VII. All limit points lying in a plane p are the limit points of planes 
of the extent which pass through the polar of p. 

VIII. For a surface of second class without singular planes it is pos 
sible in an infinite number of ways to construct self-polar tetrahedrons. 

IX. If a line m lies in a plane p, and r and s are the planes of the 
extent which pass through m, and q is the plane through m and the 
pole of p, then p and q are harmonic conjugates to r and s. 

99. Classification of surfaces of the second class. The previous 
sections enable us to write the simplest forms to which the equa 
tion of a surface of the second class may be reduced. 

I. A = 0. Since the planes envelop a surface of type I, 93, 
we may take the results of that section and find the plane equation 
corresponding to each type there. Consequently, if no account is 
taken of real values the equation of the plane extent may be 
written as ? + +*. += 0. 



SURFACES OF SECOND ORDER AND SECOND CLASS 239 

If the coefficients in the original equation are real and the origi 
nal coordinates are also real, then, by a real change of coordinates, 
the equation takes one or another of the forms 

ul -f ul + ul 4- ul = 0, 



II. A = 0, but not all the first minors are zero. We have already 
obtained equation (6), 97, as a possible equation in this case. 
If no account is taken of reals this equation can be reduced to 
the form 



In the domain of reals there are two types : 
1. Planes tangent to a real plane curve 



2. Planes tangent to an imaginary plane curve 

<+<+<=0. 

III. A = 0, all the first minors are zero but not all the second 
minors are zero. As shown in 97, the equation can be reduced 
to the single type ^ + ^ = Q 

if no account is taken of reals, and to the following two types in 
the domain of reals : 

1. Two real bundles of planes 

u*-u2=0. 

2. Two imaginary bundles of planes 

tt>+ic*sO. 

IV. A = 0, all the first and second minors are zero. As shown 
in 97, there is here only one type of equation, 

<=o, 

representing a double bundle of planes. 






CHAPTER XIV 

TRANSFORMATIONS 

100. Collineations. A collineation in space is a point transforma 
tion expressed by the equations 



px f 2 = 



We shall consider only the case in which the determinant \a ik \ 
is not zero, these being the nonsingular collineations. Then to any 
point x { corresponds a point x(, for the right-hand members of (1) 
cannot simultaneously vanish. Also to any point x[ corresponds 
a point x i given by the equations obtained by solving (1), 



(7x i = A u x[ + A 2i x 2 + A si x a + AI& , (2) 

where, as usual, A ik is the cofactor of a ik in the expansion of the 
determinant |%|. 

By means of (1) any point which lies on a plane with coordi 
nates u { is transformed into a point which lies on a plane with 
coordinates w, where 



t (3) 

and <ju = + 



The following theorems, similar to those of 40, may be proved 
by the same methods there employed. 

/. By a nonsingular collineation points, planes, and straight lines 
are transformed into points, planes, and straight lines respectively in 
a one-to-one manner. 

II. The nonsingular collineations form a group. 

III. If P l , P 2 , P%, PI, and P b are five arbitrarily assumed points no 
four of which lie in the same plane, and 1^, P 2 , P%, P, and P^ are also 

240 



TRANSFORMATIONS 241 

five arbitrarily assumed points no four of which lie in the same plane, 
there exists one and only one collineation by means of which P^ is trans 
formed into Pj, P 2 into P 2 , P s into P s f , P into JJ , and P b into P 5 r . 

IV. A nonsingular collineation establishes a projectivity between the 
points of two corresponding ranges or the planes of two corresponding 
pencils, and any such projectivity may be established in an infinite 
number of ways by a nonsingular collineation. 

V. Any two planes which correspond by means of a nonsingular 
collineation are protectively transformed into each other. 

101. Types of nonsingular collineations. A collineation has a 
fixed point when x[= x { in the equations (1), 100. Fixed points 
are therefore given by the equations 



a * x i + ^2 + Va + <>44 - P) x = - 

The necessary and sufficient conditions that these equations 
have a solution is that p satisfies the equation 



Similar conditions hold for the fixed planes. By reasoning 
analogous to that used in 41 we may establish the results : 

Every collineation has as many distinct fixed planes as fixed points, 
as many pencils of fixed planes as lines of fixed points, and as many 
bundles of fixed planes as planes of fixed points. 

In every fixed plane lie at least one fixed point and one fixed line, 
through every fixed line goes at least one fixed plane, on every fixed 
line lies at least one fixed point, through every fixed point go at least 
one fixed line and one fixed plane. 

With the aid of these theorems we may now classify the 
collineations. For brevity we shall omit much of the details of the 
work, which is similar to that of 41.* In the following equations 

* As in 41, the use of Weierstrass s elementary divisors would simplify the work. 
See footnote, p. 86. 



242 THREE-DIMENSIONAL GEOMETRY 

the letters a, b, c, d represent quantities which are distinct from 
each other and from zero. 

A. At least four distinct fixed points not in the same plane. The 
four points may be taken as the vertices of the tetrahedron of 
reference A BCD (see Fig. 52, 82). We have, then, the following 
types : 

TYPE I. px( = ax v 



The collineation has the isolated fixed points A, B, C, D, and the 
isolated fixed planes ABC, BCD, CD A, DAB. 

TYPE II. px( = ax r 

px f 2 = ax 2 , 
px s = cx s , 

px( = dx. 

The collineation has the isolated fixed points A, B, the line of 
fixed points CD, the isolated fixed planes ACD, BCD, and the 
pencil of fixed planes with axis AB. 

TYPE III. px l = ax v 



px = ex,. 

The collineation has the two lines of fixed points AB, CD and 
the two pencils of fixed planes with the axes AB, CD. 

TYPE IV. px( = ctXf 

P4 = ax * 
px z = ax s , 

px( = dx 4 . 

The collineation has the isolated fixed point A, the plane of 
fixed points BCD, the isolated fixed plane BCD, and the bundle of 
fixed planes with vertex A. 



TRANSFORMATIONS 243 

TYPE V. px( = a,Xf 



px f 2 = ax z , 



All points and planes are fixed. It is the identical transformation. 

B. At least three distinct fixed points not in the same straight line and 
no others not in the same plane. The fixed points may be taken as the 
points A, B, D. There are three fixed planes, one of which is ABD, 
and the others must intersect ABD in one of the three fixed lines 
AB, CD, DA. We may take one of these planes as DBC(x^= 0). 
Then in that plane we have a collineation in which B and D are 
the only fixed points. By proper choice of the vertex C the collinea- 
tions in the plane # 4 = may be given the forms found in 41. 
Hence for the space collineations we find the following types : 

TYPE VI. px[ = ax l + x 2 , 



The collineation has the isolated fixed points A, B, D and the 
isolated fixed planes ABD, ADC, BCD. 
TYPE VII. px[ = ax l + x 2 , 



The collineation has an isolated fixed point D, a line of fixed 
points AB, the isolated fixed plane ABD, and the pencil of fixed 
planes with the axis CD. 

TYPE VIII. px( = ax l + x 2 , 



px[ = dx^. 

The collineation has the isolated fixed point A, the line of fixed 
points BD, the isolated fixed plane BCD, and the pencil of fixed 
planes with the vertex AD. 



244 THEEE-DIMENSIONAL GEOMETRY 

Type VIII is distinguished geometrically from Type VII by the 
fact that in Type VIII the line of fixed points intersects the axis 
of the pencil of fixed planes and in Type VII this is not the case. 

TYPE IX. px( = ax l + x^ 

p4 = ax ? 

P4 = ax sJ 

px[ = ax 4 . 

The collineation has the plane of fixed points ABD and the 
bundle of fixed planes with vertex D. 

C. At least two distinct fixed points and no others not in the same 
straight line. The fixed points may be taken as B and D. There 
must be two distinct fixed planes of which one must pass through 
BD and the other may. There are two subcases each leading to 
two types of collineations. 

1. If both fixed planes pass through BD they may be taken as 
x z = and x^= 0. Then in each of these planes we have a collineation 
of Type IV or Type V of 41. By proper choice of the points A and 
C we have, accordingly, the following types of space collineations : 

TYPE X. px[ = ax l + x^ 

P4 = *v 

px 3 = bx s + x^ 

rt= K- 

The collineation has the isolated fixed points B, D and the isolated 
fixed planes ABD, BCD. 

TYPE XI. px[ = ax l + x 2 , 

px 2 = ax^ 



The collineation has the line of fixed points BD and the pencil 
of fixed planes with the axis BD. 

2. If only one of the fixed planes passes through BD the other 
must contain one of the fixed points B or D. In this case we may 



TRANSFORMATIONS 245 

take the two fixed planes as x 4 = and # 3 = 0. Then in the plane 
BCD we have a collineation of Type IV or Type V of 41 and 
in ABD one of Type VI of 41. By proper choice of the points 
C and A, therefore, we have the following types : 

TYPE XII. px(=ax l + x v 

px 2 = ax 2 + x 

px( = bx 3 , 

px 4 = ax,. 

The collineation has the fixed points B, D and the fixed planes 
BCD, ADC. 

TYPE XIII. x=ax + x., 



The collineation has the line of fixed points BD and the pencil 
of fixed planes with the axis DC. 

D. Only one fixed point. The fixed point may be taken as D. 
The fixed plane which must exist may be taken as x 4 = 0. Then 
in that plane the collineation is of Type VI, 41, and the points 
C and B may be so chosen that the equations take the form of 
Type VI there given. To do this we first select x a = Q, x 4 =0 as the 
fixed line in the plang x 4 = 0. The point A may be taken as any 
point outside of # 4 = (^, If A is the point into which A is trans 
formed, the line A A may be taken as x^ 0, x^= 0. This fixes the 
point B. Then C is determined, as in Type VI, 41. The result 
is the following type: 

TYPE XIV. px[=ax 1 + x zr 



px(= ax,. 

The above types exhaust the cases of a nonsingular collineation. 
In a singular collineation there exist exceptional points, lines, or 
planes. The discussion of these is left to the student. 



246 THREE-DIMENSIONAL GEOMETRY 

EXERCISES 

1. Considering the translation 

x =x + a, y =y + b, z =z + c 

as a collineation, determine its fixed points and the type to which it 
belongs. 

2. Considering the rotation 

x = x cos <j> y sin <, ?/ = x sin< + y cos <, = 

as a collineation, determine its fixed points and the type to which it 
belongs. 

3. Considering the screw motion 

x = x cos (f> y sin <, y = x sin <j> + y cos <, z =kz 

as a collineation, determine its fixed points and the type to which it 
belongs. 

4. Set up the formulas for the singular collineation known as 
" painter s perspective," by which any point P is transformed into that 
point of a fixed plane p in which the line through P and a fixed point 
meets p. 

5. Find all possible types of nonsingular collineations. 

102. Correlations. A correlation of point and plane in space is 
defined by the equations 

pu( = a {l x^ + a i2 x. 2 + a. B x s + a i4 x , ( = 1,2, 3, 4) (1) 



where u { are plane coordinates and x i are point coordinates. The 
correlation is nonsingular when |%|^ 0, and we shall consider only 
such correlations. Then any point x i is transformed into a definite 
plane w t , and any plane u\ is the transformed element of a definite 
point, so that the correspondence of an element and its transformed 
element is one-to-one. The points x i which lie on a plane with 
coordinates u { are transformed into planes u[ which pass through a 

point xL where ~ N 

px\= A^ + A i2 u z + A i3 u s + A^, (^) 

where A ik is the cof actor of % in the determinant | a ik |. We may 
say, therefore, that the plane u { is transformed into the point x[. 
Points which lie on a line I are transformed into planes through a 
line Z , so that we may say that the line I is transformed into the 
line I . 



TRANSFORMATIONS 247 

If the point P(x^) is transformed into the plane p (u{), then, by the 
same operation, the plane p f is transformed into the point P"(#"), 
where, from (2), 

px( = 4X + 4X + 4X + 4X- 

The last equations solved for u[ give 

pu\ = a^xC + 2 ,4 + a si x s + a 4i arj - (4) 

The points x f and rrj are in general distinct. That they should 
coincide it is necessary and sufficient, as is seen by comparison 
of (1) and (4), that 

J X i + Oia - /^2i) x * + Oia - PV) ^3 + Ou ~ ? a 4i) *4 = 



where p must satisfy the condition 

W a* a* ~ ^ ^ 

U ~~ ^^14 ^43 "" /^24 ^43 "" ^34 ^44 ~~ ^44 

in order that equations (5) may have a solution. 

When the coordinates of a point P satisfy equations (5), it and 
the plane p\ into which it is transformed, form a double pair of the 
correlation. Since (6) is of the fourth degree we see that in general 
a correlation has four double pairs, but may have more. 

The double pairs may be made the basis of a classification of 
correlations, as was done in the case of the plane, but we will not 
take the space to do so. Of special interest is the case in which each 
point of space is a point of a double pair. For this it is necessary 
and sufficient that equations (5) should be satisfied for all values 
of x. This can happen in only two cases : 

1. p = l j a ki = a ;k . 2. p = - 1, ..= 0, a ki = - a ik . 

In the first case the correlation is evidently a polarity with 
respect to the conic ^^/^ t ^ 0, and by proper choice of coordi 
nates it may be represented by the equations 



248 THREE-DIMENSIONAL GEOMETRY 

In the second case the correlation has the form 



and represents a null system, which will be discussed later. It will 
be shown that by choice of axes the correlation may be reduced 
to the standard form 



_ 





Another question of interest is to determine the condition under 
which a point P lies in the plane p , into which it is transformed. 
From equations (1) it follows at once that the coordinates of P 
must satisfy the equation 



This equation is satisfied identically only in the case of the null 
system ; otherwise it determines a quadric surface K^ the locus of 
the points P which lie in their respective transformed planes. 
Similarly, the planes p which pass through their respective trans 
formed points envelop the quadric K^ 



which is in general distinct from K^ 

EXERCISES 

1. Prove that if P and p are a double pair the plane p is the polar 
plane of P with respect to the conic K^ 

2. Prove that a correlation is an involutory transformation only in 
the case of a polarity or a null system. 

3. Explain why there is no analog of the null system in plane 
geometry. 

4. Prove that any correlation is the product of a collineation and a 
polarity. 



TRANSFORMATIONS 249 

103. The projective and the metrical groups. The product of 
two nonsingular collineations or of two nonsingular correlations 
is a nonsingular collineation. Hence the totality of all collineations 
and correlations form a group, since this totality contains the 
identical substitution. Projective geometry may be defined as that 
geometry which is concerned with the properties of figures which 
are invariant under the projective group. In this geometry the 
plane at infinity has no unique property distinct from those of 
other planes nor is the imaginary circle at infinity essentially 
different from any other conic, and all questions of measurement 
disappear. Quadric surfaces are distinguished only by the presence 
and nature of their singular points. 

Subgroups exist in great abundance in the group of projections. 
For example, the collineations taken without the correlations form 
a subgroup, but the correlations alone form no group. All colline 
ations with the same fixed points obviously form a subgroup. 
Again, all collineations which leave a given quadric surface inva 
riant form a subgroup. Of great importance among these latter 
is the group which leaves the imaginary circle at infinity invariant. 
This is the metrical group, which leaves angles invariant and multi 
plies all distances by the same constant. 

The general form of a transformation of the metrical group is 

px r = l^x + mjj + n 



pz = l z 
pt = f, 
where the coefficients satisfy the conditions 

I? + y + *3 2 = < + < + ml = n* + n* + w, a , (2) 

l^ + I 2 m 2 + l s m s = m^ + mri z + w 8 w g = n^ + n^ + n^ = 0. (3) 

It is easy to see that the distance between two transformed 
points is by this transformation k times the distance between the 
original points, where k 2 is the common value of the expressions 
in (2), and, conversely, that a collineation which multiplies all 
distances by the same constant is of the form (1). The preser 
vation of angles follows from elementary theorems on similar 
triangles. 



250 THREE-DIMENSIONAL GEOMETRY 

All transformations of the metrical group which leave a plane p 
fixed form a group of collineations in that plane by which the 
circular points at infinity are invariant. This group is therefore 
the metrical group in p, and the projective definitions of angle 
and distance given in 50 stand. 

EXERCISES 

1. If D is the determinant of the coefficients I, m, n in (1), show that 
D = k s . 

2. Show that the necessary and sufficient condition that (1) should 
represent a mechanical motion is that D = -j- 1, and that it should repre 
sent a motion comhined with a reflection on any plane is that D = 1. 

3. Show that if D = 1 in addition to conditions (2) and (3), we have 

1} + mf + n? = II + ml + n} = Ij + mj + n} = 1, 
IJt 4- w^ + n^ = I 2 l s 4- ra 2 ra 3 + nji s = l^ + m^ + n^ = 0. 

104. Projective geometry on a quadric surface. It has already 
been noted (69) that the geometry on a surface of second order 
with the use of quadriplanar coordinates is dualistic to the geom 
etry on the plane with the use of tetracyclical coordinates. For in 
each case we have a point defined by the ratios of four quantities 
x \"> x v x ^ x ^ bound by a quadratic relation 

(*)=0, (1) 

which is, on the one hand, the equation of the quadric surface 
and, on the other hand, the fundamental relation connecting the 
tetracyclical coordinates. 

Any point / on the quadric surface may be taken as correspond 
ing to the point at infinity on the plane, since the point at infinity 
is in no way special in the analysis. Any linear equation 

2><*,-= (2) 

represents a plane section of the surface or a circle on the plane. 
Should the section pass through /, the circle on the plane becomes 
a straight line, but circles and straight lines have no analytic 
distinction in this geometry. 

If y i is a point on the quadric surface and we have, in (2), 

(3) 



TRANSFORMATIONS 251 

the plane (2) is tangent to the surface, and the circle on the plane 
is a point circle. The point of tangency on the surface corresponds 
to the center of the point circle on the plane. The intersection of 
the tangent plane with the quadric surface consists of two gen 
erators. In a corresponding manner the point circle on the plane 
consists of two one-dimensional extents. Neither alone, however, 
can be represented by a linear equation in a;., and therefore they 
are not straight lines on the plane. If this is obscure it is to be 
remembered that imaginary straight lines are not -defined by any 
geometric property, but by an analytic equation. 

The intersection with the quadric surface of the tangent plane 
at / corresponds to the locus at infinity on the plane. 

The center y i of a point circle on the plane, or the point of tan 
gency of a tangent plane to the surface, is found by solving (3) 
for y r The values of y i must satisfy (1), and the substitution 
gives the equation ^ (fl) = Qj (4) 

which is the condition that a circle on the plane with tetracyclical 
coordinates should be a point circle, or that a plane in space should 
be tangent to the point circle. It is in fact simply the equation in 
plane coordinates of the quadric surface (1). 

Two circles on the plane are perpendicular when 



2 = l (,*) = <>. . (5) 

In space the pole of the plane ^a^= with respect to the sur- 

o 

face with the plane equation (4) is ft^jpj and equation (5) is 

the condition that this pole lie in the plane ]T^= 0. Hence two 
orthogonal circles on the plane with tetracyclical coordinates cor 
respond to two plane sections of the quadric surface such that 
each plane contains the pole of the other. 

A linear substitution of the tetracyclical coordinates corresponds 
to a collineation in space which leaves the quadric surface invariant. 
The geometry of inversion on the plane is therefore dualistic to the 
geometry on the quadric surface which is invariant with respect to 
collineations which leave the surface unchanged. Two points on 
the plane which are inverse with respect to a circle C correspond 
to two points on the quadric surface such that any plane through 



252 THKEE-DIMENSIONAL GEOMETRY 

them passes through the pole of the plane corresponding to C or, 
in other words, such that the line connecting them passes through 
the pole of the plane corresponding to C. Since the center of a 
circle on the plane is the inverse of the point at infinity with 
respect to that circle, the point on the quadric which corresponds 
to the center of a circle may be found by connecting the point / 
with the pole of the plane corresponding to the circle. 

An inversion with respect to a circle corresponds in space to a 
collineation which transforms each point into its inverse with 
respect to a fixed plane. That is, if the fixed circle corresponds to 
the intersection of the quadric with a plane M, and K is the pole 
of M, an inversion with respect to M transforms any point P on 
the quadric into the point P^ where the line KP^ again meets the 
quadric. The collineation which carries out this transformation 
has the plane M as a plane of fixed points and the point K as a 
point of fixed planes. 

Consider now the parameters (X, fi) on the surface, defined as in 
96. They may be taken as the coordinates of a point on the sur 
face and may be interpreted dualistically to the special coordinates 
of 70. The two families of generators are then dualistic to the two 
systems of special lines of 70, and the locus at infinity on the plane 
is dualistic to the generators through the point / of the surface. 

The bilinear equation 

d^X/i + a 2 \ + a 8 /Lt + a^ = (6) 

represents a plane section of the quadric surface and is dualistic 
to the equilateral hyperbola on the plane with two special lines as 
asymptotes. A section of the quadric surface through / corresponds 
to an ordinary line on the plane, from which it is evident that by 
the use of the special coordinates the straight line has the properties 
of the equilateral hyperbola. 

Any collineation of space which leaves the quadric surface inva 
riant gives a linear transformation of X and of /JL. This is evident 
from the fact that the collineation must transform the lines of the 
surface into themselves in a one-to-one manner. It may also be 
proved analytically from the relations of 96. 

Conversely, any linear substitution of X and /u. corresponds to a 
collineation which leaves the quadric invariant. 



TRANSFORMATIONS 253 

Consider in fact the substitution 



which leaves the generators of the second family fixed and trans 
forms the generators of the first family. From (4), 96, it is easy 
to compute that this is equivalent to the collineation 
px, = (a + 8X + i(a - 8X+ (7 - Xs- id/3 + 



8 
= (!3 - 7X- * ( + 7X + 0* 



^ 4 = t 
Similar results can be obtained for the transformation 



by which the generators of the first family are fixed, and for the 
product of (7) and (9). 

Finally, the collineation corresponding to the transformation 

- - 



by which generators of the two families are interchanged, is easily 

computed. 

EXERCISES 

1. Show that if the quadric (1), 96, is the sphere x 2 + if + = 1, 
the transformation X = e *X , //, = e**/* represents a rotation of the sphere 
about the axis OZ through an angle <. 

2. Show that the transformation A.= /x f , /JL=\ replaces each 
point of the sphere of Ex. 1 by its diametrically opposite point. 

3. Obtain a transformation of X, p which represents a general rota 
tion of the sphere in Ex. 1 about any axis through its center. 

105. Protective measurement. The definition of projective meas 
urement, given in 47 for the plane, can evidently be generalized for 
space, and only a concise statement of essentials is necessary here. 

Let o>(a;)=0 (1) 

be the equation of any quadric surface taken as the fundamental 
quadric for the measurement, and let 

n<=o (2) 

be the equation of the same surface in plane coordinates. 



254 THREE-DIMENSIONAL GEOMETRY 

If A and B are any two points and T l and T 2 are the points in 
which the line AB meets the quadric, then the distance D between 
A and B is defined by the equation 



or if y i and z. are the coordinates of A and B respectively, 

OOi 





(y. *) - [ (y, *)] 2 - [ 

Also, if a and b are two planes and ^ and 2 are the two tan 
gent planes to the quadric through the intersection of a and 5, the 
angle < between a and b is defined by the equation 



= 



where w t - and v. are the coordinates of a and 5 respectively. 

Two planes are perpendicular if each passes through the pole of 

the other ; for, in (4), if fl (u, v) = 0, then <j> = l - log ( 1) = - + mr. 

A A. 

A line is perpendicular to a plane p if every plane through the 
line is perpendicular to p ; that is, if the line passes through the 
pole of p. 

We may define the angle between two lines in the same plane 
as the angle between the two planes through the lines and perpen 
dicular to the plane of the lines. That is the same as defining the 

y 

angle between the two lines as - times the logarithm of the cross 

ratio of the two lines and the two tangent lines drawn in their plane 
to the quadric surface. 

Any plane cuts the quadric surface in a conic, and the definition 
of angle and distance is the same as in the projective measurement 
of 47, in which this conic is the fundamental one. Projective 
plane measurement is therefore obtained by a plane section of 
projective space measurement. 

As in Chapter VII we have three cases : 

I. The hyperbolic case. The fundamental quadric is real, and we 
consider only the space inside of it. The geometry in the plane is 
the same as in 48. 



TRANSFORMATIONS 255 

II. The elliptic case. The fundamental quadric is imaginary. 
The geometry in the plane is the same as in 49. 

III. The parabolic case. The fundamental quadric in plane coor 
dinates may be taken as 



which is that of a plane extent consisting of planes tangent to a 
conic in the plane # 4 = 0. If this conic is the circle at infinity, the 
measurement becomes Euclidean. 

If the conic is a real circle at infinity, for example the circle 



we have a measurement in which 



and the angle between the two planes 

ax + by + cz + di=Q and a x + b y -f c z + d t = 

aa +W cc 
is given by cos 6 = . 

O / / rt 7 n 



Through any point in space goes a real cone, such that the dis 
tances from its vertex to points inside it are imaginary, distances 
from its vertex to points outside it are real, and distances from its 
vertex to points on it are zero. Any plane section through the 
vertex is divided into regions with the properties described in 50. 

106. Clifford parallels. When a system of projective measure 
ment has been established, the concept of parallel lines may be 
introduced by adopting some property of parallel lines in Euclidean 
geometry as a definition. Perhaps the most obvious as well as the 
most common definition is that parallel lines are those which in 
tersect at infinity. By this definition, in parabolic space one and 
only one line can be drawn through a point parallel to a given line, 
in hyperbolic space two such parallels can be drawn, and in elliptic 
space no real parallel can be drawn. 

In elliptic space, however, there exist certain real lines called 
Clifford parallels which have other properties of parallel lines as they 
exist in Euclidean space. We will proceed to discuss these lines. 

We have seen that any linear transformation of the parameters 
X and /JL which define a point on a quadric surface correspond to 



256 THREE-DIMENSIONAL GEOMETRY 

a collineation which leaves the quadric invariant. Among these 
transformations are those of the type 



>-< 



which transform the generators of the first family among themselves 
but leave each generator of the second family unchanged. 

For reasons to be given later we call such a transformation a 
translation of the first kind. 

Similarly, the transformation 

. . , mp +.n , ON 

X = X , fi = -^- , (2) 

Pff+t 

by which the generators of the second family are transformed but 
each of the first family is left unchanged, is called a translation of 
the second kind. 

Consider a translation of the first kind. On the fundamental 
quadric any generator of the second family is left unchanged as a 
whole, but its individual points are transformed, except two fixed 
points, for which _ aX+ff 

- 7 x + s* 

This equation defines two generators of the first kind, all of 
whose points are fixed. Hence, in a translation of the first kind there 
are, in general, two generators of the first kind which are fixed point 
by point. We say "in general" because it is possible that the two 
roots of (3) may be equal. 

Call the two fixed generators g and h. Then any line which in 
tersects g and h is fixed, since two of its points are fixed. Also 
through any point P in space one and only one line can be drawn 
intersecting g and h. Therefore, any point P is transformed into 
another point on the line which passes through P and intersects g and h. 

Since we are dealing with a case of elliptic measurement the lines 
g and h are imaginary. Then, if a real point P is transformed into 
another real point, the roots of (3) must be conjugate imaginary, 
since a real line intersects an imaginary quadric whose equation has 
real .coefficients in conjugate imaginary points corresponding to con 
jugate imaginary values of X and p. Therefore, if a translation of the 
first kind transforms real points into real points, there must be two dis 
tinct fixed generators corresponding to conjugate imaginary values of X. 



TBANSFORMATIONS 257 

This may also be established by equations (8), 104. That these 
may represent a real substitution 8 must be conjugate imaginary to 
#, and 7 conjugate imaginary to /3. We therefore place a = d 4- ic, 
B = d ic, fi = b 4- ia, 7 = 64- ia, and have 

px^ = dx( cx 2 4- bx f s 4- ^ , 
px, = cx( 4- dx[- ax z 4- ^1 , 
/o^ 3 = fa J 4- ax 2 4- ^3 4- raj , 
px = ax[ bx . 2 cx r 3 4- dx[. 

With these values of #, /8, 7, and 8 the roots of (3) are conjugate 
imaginary. 

To find the projective distance between a point x i and its trans 
formed point x[, we use equations (4) and substitute in (3), 105, 

placing K=- There results 



d 



D = - log = cos" 

which is a constant. Hence, by a translation of the first kind each 
point of space is moved through a constant projective distance on the 
straight line which passes through the point and meets the two fixed 
generators on the fundamental quadric. 

It is this property which gives to the transformation the name 
" translation " and to the lines which intersect the two fixed gen 
erators the name " parallels." By the transformation the points of 
space are moved along the Clifford parallels in a manner analo 
gous to that in which points are moved along Euclidean parallels 
by a Euclidean translation. 

In the projective space a dualistic property exists. Since the 
Clifford parallels are fixed, any plane through one of them is trans 
formed into another plane through it. Now any plane contains one 
Clifford parallel, since it intersects each of the fixed generators in 
one point. If u. and u[ are the original and the transformed plane 
respectively, the angle between them is, by (4), 105, 



258 THREE-DIMENSIONAL GEOMETRY 

Hence, by a translation of the first kind each plane of space is turned 
about the Clifford parallel in it through a constant angle which is equal 
to the distance through which points of the space are moved. 

Similar theorems hold for translations of the second kind. The 
two kinds of translations differ, however, in the sense in which the 
turning of the planes takes place. 

By a translation of the second kind Clifford parallels of the first 
kind are transformed into themselves. For by the translation of 
the second kind all generators of the first kind are fixed, and conse 
quently any line intersecting two such generators is transformed into 
a line intersecting the same two generators. Hence two Clifford 
parallels are everywhere equidistant if the distance is measured on 
Clifford parallels of the other kind. 

Let LK and MN be two Clifford parallels of the first kind, g 
and h the two fixed generators which determine the parallels, and 
PQ any line intersecting both LK and MN. The line PQ intersects 
two generators g f and h of the second kind and is therefore one 
of a set of Clifford parallels of the second kind. Therefore there 
exists a transformation of the second kind by which PQ is fixed 
and LK is transformed into MN, P falling on Q. Hence the 
angles under which PQ cuts LK and JOT -are equal, of course in 
the projective sense. That is, if a line cuts two Clifford parallels, 
the corresponding angles are equal. 

In particular the line may be so drawn as to make the angle 
LPQ a right angle. For if Q is on MN, the point Q and the line 
LK determine a plane p, and in this plane a perpendicular can be 
drawn from Q to LK. To do this it is only necessary to connect Q 
with the point in which the plane p is met by the reciprocal polar 
of LK with respect to the quadric surface. 

Hence, from any point in one of two Clifford parallels a common 
perpendicular can be drawn to the two, and the portion of the perpen 
dicular included between the two parallels is of constant length. 

107. Contact transformations. A transformation in space, expres 
sible by means of analytic relations between the coordinates of 
points, may be of three kinds according as points are transformed 
into points, surfaces, or curves respectively. We shall find it con 
venient to employ Cartesian coordinates in discussing these trans 
formations and to introduce the concept of a plane element. 



TRANSFORMATIONS 259 

Let (#, y, z) be a point in space and let Z z =p(Xx) + q(Y y) 
be a plane through it. Then the five variables (#, y, z, p, q) define 
a plane element, which may be visualized as an infinitesimal portion 
of a plane surrounding a point. In fact, not the magnitude of the 
plane but simply its orientation comes into question, just as, in 
fixing a point, position and not magnitude is considered. If any 
one of the five elements is complex, then the plane element is 
simply a name for the set of variables (x, y, z, p, q). 

Since the five variables are independent, there are oo 5 plane ele 
ments in space. Of chief interest, however, are two-dimensional 
extents of plane elements. Such an extent we shall denote by M 2 
and shall consider three types: 

1. Let the points of the plane elements be taken in the surface 
z =f(x, y) and let p and q be determined by the equations 

3 ^ 

p = -> q = - More generally, let #, y, and z be defined as 
ex cy 

functions of two variables u and v, and let p and q be determined 

by the equation . 

dz=pdx + qdy (1) 

for all differentials du and dv. Then 

dz dx dii 

r = p + q^ 

du cu du 

dz dx dy . 



whence p and q are also determined as functions of u and v. 

In either definition the M z consists of the plane elements 
formed by the points of a surface and the tangent planes at 
those points. 

2. Let the points of the plane elements be taken as functions 
of a single variable u and let p and q be again determined by 
equation (1), where one of the two (say p) is arbitrary and the 
other (say q) is thus determined in terms of p and u. The M Z 
then consists of the points of a curve and the tangent planes to 
the curve at those points. The points themselves form a one- 
dimensional extent, and through each point goes a one-dimensional 
extent of planes ; namely, the pencil of planes through the tangent 
line to the curve. 



260 THREE-DIMENSIONAL GEOMETRY 

3. Let (x, y, z) be a fixed point and let p and q be arbitrary and 
independent. The M 2 then consists of a point with the bundle of 
planes through it. In this case, also, equation (1) is true, since 
dx, dy, and dz are all zero. 

It is clear that the M 2 s defined above do not exhaust all pos 
sible types of two-dimensional extents of plane elements. For 
example, we might take the points as points on a surface and the 
planes as uniquely determined at each point but not tangent to 
the surface ; and other examples will occur to the student. The 
above-mentioned types exhaust all cases, however, for which equa 
tion (1) is true, as the student may verify. We shall say that a set 
of plane elements satisfying (1) form a union of elements. 

Two M^s are said to be in contact when they have a plane 
element in common. From this definition two surfaces, or a curve 
and a surface, are in contact when they are tangent in the ordi 
nary sense, a point is in contact with a surface or a curve when 
it lies on the surface or the curve, two curves are in contact when 
they intersect, and two points are in contact when they coincide. 

A contact transformation is a transformation by which two M^s 
in contact are transformed into two .3f 2 s in contact. There are 
three types of such transformations, which we shall proceed to 
discuss in the following sections. 

108. Point-point transformations. This transformation is defined 
by three equations of the form 



(1) 



or, more generally, F^(x, y-, z, x , y , z 1 ) = 0, 

I\(x,y,z,x ,y ,z )=Q, (2) 

F 9 (x, y, z, x , /, O=0, 

where we make the hypothesis that equations (1) can be solved 
for x, y, z and equations (2) for z, /, z and a/, y\ z , and that all 
functions are continuous and may be differentiated. Within a prop 
erly restricted region the relations between x, y, z and a/, y , z are 
one to one, a point goes into a point, a surface into a surface, 
and a curve into a curve. 



TRANSFORMATIONS 



261 



A direction dx:dy: dz is transformed into a direction dx i dy : dz , 



where 



, , dx . dx 
dx = dx + dy 

dx dy 



dx cy 

dz dz 
dz = ax H a y 

dx cy l 



d 
dz Z * 

~fo dZ 

dz 

-dz. 

dz 



(3) 



From this it follows that two tangent surfaces are transformed 
into tangent surfaces. More specifically, the relation 



dz = p dx + q dy, 
which defines a union of line elements, is transformed into 



(4) 



dz 
dx 
dy 



dz 



dz dz 



dx +P dz 

dx dx dx 



dz 



dx 
dj/_ 

ix~ 



+P dz 



dz 



dx 
dy q ~dz 



= 0. 



(5) 



If now we define p and q so that this relation is 
dz = p dx +q dy , 



(6) 



a union of plane elements (x, y, z, p, q) is transformed into a union 
of plane elements (V, y , z , p , q ). From equations (5) and (6), 



These equations adjoined to (1) form, together with (1), the 
enlarged point transformations. 

A collineation is an example of a point transformation. Another 
example of importance is the transformation by reciprocal radius, 
or inversion with respect to a sphere. If the sphere has its center 
at the origin and radius &, the transformation is 

,_ ** 



262 THREE-DIMENSIONAL GEOMETRY 

EXERCISE 

Discuss the properties of the inversion with respect to a sphere, 
especially with reference to singular points and lines. 

109. Point-surface transformations. Such a transformation is 
denned by the equation 

f(x,y,z,xi,y ,z>)=$, (1) 

with the usual hypotheses of continuity and differentiability of /. 
An example is a correlation since it may be expressed by the single 
equation 

(a n x + a^y + a^z -h M ) x + (a^x + a^y + a^z + aj y 

+ OV+ a & + v + a J z + a 4i x + a ^y + v + a u = - 

By equation (1), if (z, j/, 2) is fixed, (V, y, 2 ) lies on a surface m , 
and we say a point P is transformed into a surface m f . If P f (a/, ?/ , 2 ) 
is fixed, the point (x, y, z) describes a surface w, where the surfaces 
m and w are not necessarily of the same character. If P is on m! it is 
obvious that m contains P. In other words, if P describes a surface 
w, the corresponding surface, m , continues to pass through P . We 
say, therefore, that the surface m is transformed into a point P . 

If P describes any surface S (differing from an m surface), the 
surface m will in general envelop a surface S , the transformed 
surface of S. Analytically, from the general theory of envelopes, if 
the equation of S is z = 



and p = ^ q = -, the equation of S f is found by eliminating x, y, 
and 2 from (1) and (2) and the two equations 

2-P 2-F 

\ = 0, (3) 

- = 0. (4) 

vy vz 

Furthermore, the tangent plane to S at any point is the same as 
the tangent plane to m at that point, and hence, if we use p and q f to 
fix that plane, we have / / 



TRANSFORMATIONS 263 

We now have five equations, namely (1), (3), (4), (5), and (6), 
establishing a relation between a plane element (#, y, z, p, q) and 
a plane element (V, y, 2 , //, q ). These equations may be solved 
to obtain the form 



. 

X = 



, z,p, q), 



p =<>*?, y, *, p, ? 
^=* 6 (^y *.? ?)> 

which form the enlarged point-surface contact transformation. 

EXERCISES 

1. Study the transformation defined by the equation 

x * + 2/ 2 + 2 (* 4- 2/2/ + ) = 0. 

2. Study the transformation denned by the equation 

(x - x f + (y - y) 2 + (* - * ) 2 = a\ 

110. Point-curve transformations. Consider a transformation 
defined by the two equations 

f^x, y, z, x ,y , z f )=Q, 
f 2 (x,y,z,x ,y ,z )=0. 

If a point P (z, y, 2) is fixed, the locus of P 1 (V, / , 2 ) is a 
curve k defined by equations (1). Similarly, if P is fixed, the 
locus of P is a curve k. Hence the transformation changes points 
into curves. 

If P describes a curve (7, the curve k takes oo 1 positions and in 
general generates a surface. The oo 1 curves k f may, however, have 
an envelope C", which is then the transformed curve of C. Or, 
finally, if C is a curve &, the corresponding curves k 1 pass through 
a point P , which we have seen to correspond to k. 

If the point P describes a surface $, the corresponding curves k 
form a two-parameter family of curves. The envelope of the family 
is a surface S which corresponds to S. 

To work analytically let us form from (1) the equation 

/,+v;=o. 



264 THREE-DIMENSIONAL GEOMETRY 

With (V, y , z ) fixed, (2) represents a pencil of surfaces through 
a &-curve, and the tangent plane to any one of these surfaces at a 
point on the &-curve has a p and a q given by the equations 



a* 



a? 

There is therefore thus defined a pencil of plane elements through 
a point P and tangent to a &-eurve through that point. 

Similarly, with (x, y, z) fixed, equation (2) defines a pencil of 
surfaces through a & -curve, and a corresponding pencil of plane 
elements is defined by (V, y , z ) and 



From (3) and (4) it is easy to compute that dzpdx qdy is 
transformed into dz 1 p dx q dy except for a factor. So that if 
(x, y, z, p, q) is transformed into (V, y 1 , d, p\ q ) by means of (1), 
(3), and (4), a union of plane elements is transformed into a 
union of plane elements. 

From the six equations (1), (3), (4) we may eliminate X and 
obtain five equations which may be reduced to the form 



which define the enlarged point-curve contact transformation 
derived from (1). 

Consider a fixed point P(a, 6, (?) with the Jf 2 of plane elements 
through it. Equations (1) define a & -curve, and we may consider 
them solved for z and y in terms of a/. In (3) jt? and <? may be 
taken arbitrarily. Then, if the values of z and y f in terms of a? are 
substituted in (3), both X and x may be determined. Finally, 



TRANSFORMATIONS 265 

p r and <f are determined from (4). This shows that a definite 
plane element through P is transformed into a definite plane ele 
ment of a & -curve. The M 2 through P is therefore transformed 
into a M 2 along k . 

A pencil of plane elements through P will in general be trans 
formed into an M 1 of plane elements forming a strip along k , but 
if the axis of the pencil through P is tangent to a #-curve, the 
pencil will be transformed into a similar pencil at a point of the 
# -curve. 

That being established, we see that if C is any curve, and we 
take an Jf 2 of plane elements tangent to it, we shall have corre 
spondingly an M[ 2 of plane elements forming a surface. But if C 
is the envelope of ^-curves, the M 2 consists of elements tangent to 
a curve C r enveloped by ^/-curves. 

If P describes a surface S, and we take the J/ 2 of tangent ele 
ments, we shall have a corresponding Jf 2 , forming a surface S r . 
A plane element of the M 2 gives a definite plane element of a 
&-curve, as we have shown. Therefore the surface S 1 is made 
of plane elements belonging to ^ -curves and is the envelope of 
such curves. 

EXERCISE 

Study in detail the transformation defined by the equations 



CHAPTER XV 

THE SPHERE IN CARTESIAN COORDINATES 
111. Pencils of spheres. The equation 



hz + c = () (1) 

represents a sphere with the center ( > -* - ) and the radius r, 

, ,, .. \ a a a / 

given by the equation 

O / -L ,-) o -j rt 

r2 _/ +r +h 2 -ac 
a 2 

If a = 0, equation (1) represents a plane which may be regarded 
as a sphere with an infinite radius and with its center at infinity. 

For convenience we shall denote the left-hand member of equation 
(1) by S. The equation S 

shall then denote the sphere with the coefficients a,,./-, # t , 7i t -, c t . 
Consider now two spheres 

S i= 0, 2 =0. (3) 

They intersect at right angles when and only when the square 
of the distance between their centers is equal to the sum of the 
squares of their radii. The condition for this is easily found to be 



The spheres defined by the equation 

S 1 +XS 2 =0, (5) 

where X is an arbitrary parameter, form a pencil of spheres. If S l 
and S 2 are both planes, all spheres of the pencil are planes. Other 
wise the pencil contains one and only one plane, the equation of 

which is found by placing X = -- - in (5). 

a -2 

This plane, called the radical plane of the pencil, has accordingly 
the equation a.S.-a.S^O (6) 

or 



266 



THE SPHERE IN CARTESIAN COORDINATES 267 
The centers of the spheres of the pencil have the coordinates 



/ 
\ 



Xo, ttj-l-X^ flj+X 
and therefore lie in a straight line perpendicular to the radical 
plane. This line is the line of centers of the pencil. 

We have three forms of a pencil of real spheres not planes : 

1. When the spheres S l and $ 2 intersect in the same real circle C. 
The pencil consists of all spheres through C. The radical plane is 
the plane of (7, and the line of centers is perpendicular to that plane 
at the center of C. 

2. When the spheres S^ and $ 2 intersect in an imaginary circle. 
All spheres of the pencil pass through the same imaginary circle, 
but in the ordinary sense the spheres do not intersect. The radical 
plane is a real plane containing the imaginary circle, and the line 
of centers is perpendicular to it. 

3. When the spheres S l and $ 2 are tangent at a point A. The 
spheres of the pencil are all tangent at A. The radical plane is the 
common tangent plane at A, and the line of centers is perpendicular 
to the radical plane at A. 

The position of the radical plane in the second form of the pencil 
has been fixed only analytically. A useful geometrical property 
is that all the tangent lines from a fixed point of the radical plane 
to the spheres of the pencil are equal in length. For if P is 
any point of space, and M the center of a sphere of radius r, the 
square of the tangent from P to the sphere is MP r 2 . Applying 
this to a sphere of the pencil (5), we find the square of the length 
of the tangent to be . . 



1-1 i ,, -1 /, 

which can be written 

i !<! 

If the point P is in the radical plane (6), this distance is inde 
pendent of \ and hence the theorem. 

It follows from this that the radical plane is the locus of the centers 
of spheres orthogonal to all spheres of the pencil. 

Closely connected with this is the theorem : A sphere orthogonal 
to any two spheres is orthogonal to all spheres of the pencil determined 
by them and has its center on the radical plane of the pencil, 



268 THREE-DIMENSIONAL GEOMETRY 

The last part of this theorem is a consequence of the previous 
theorem. The first part is a consequence of the linear nature of 
the condition (4) for orthogonality. 

112. Bundles of spheres. The spheres defined by the equation 

S 1 +XS a +/*S 8 =0, CO 

where $ 1? 2 , S s are three spheres not belonging to the same pencil 
and X, /JL are arbitrary parameters, form a bundle of spheres. 
The centers of the spheres of the bundle have the coordinates 



/_ 



From (2) it follows that if the centers of the three spheres S t , 
$ 2 , $ 8 lie on a straight line, the centers of all spheres of the bundle 
lie on that line. The center may be anywhere on that line, and 
the radius of the sphere is then arbitrary. Hence a special case 
of a bundle of spheres consists of all spheres whose centers lie on f a 
straight line. 

More generally, if the centers of S^ $ 2 , and $ 3 are not on the 
same straight line, they will determine a plane, and the centers 

of all spheres of the bundle lie in this plane. This plane, is the 

s/s >> f e 
plane of centers, and any point in it is the center of a plan of 

the bundle. In this case the three spheres S^ $ 2 , $ 3 intersect in 
two points (real, imaginary, or coincident), and all spheres of the 
bundle pass through these points. If the two points are distinct, 
they are symmetrical with respect to the plane of centers ; if they 
are coincident, they lie in the plane of centers. Hence we see that 
a bundle of spheres consists in general of spheres whose centers lie in 
a fixed plane and which pass through a fixed point. 

The radical planes of the three spheres 15 2 , and $ 3 , taken in 
pairs, are a- a= 



which evidently intersect in a straight line called the radical axis 
of the bundle. It is perpendicular to the plane of centers and passes 
through the points common to the spheres of the bundle. The 
radical plane of any two spheres of the bundle passes through the 
radical axis. 



THE SPHERE IN CARTESIAN COORDINATES 269 

Any sphere orthogonal to three spheres of a bundle is orthogonal 
to all the spheres of the bundle because of the linear form of 
condition (4), 111. The centers of such spheres lie in the radi 
cal axis of the bundle, since by 111 they must lie in the radical 
plane of any two spheres of the bundle, and any point of the radical 
axis is the center of such a sphere. It is not difficult to show that 
these spheres form a pencil. 

In fact, to any bundle of spheres we may associate an orthogonal 
pencil of spheres and to any pencil of ~& sphered an orthogonal bundle. 
The relation of pencil and bundle is such that every sphere of the pencil 
is orthogonal to every Sphere of the bundle, the line of centers of the 
pencil is the radical axis of the bundle, and the radical plane of the 
pencil is the plane of centers of the bundle. 

As far as the details of the above theorem have not been ex 
plicitly proved in the foregoing, the proofs are easily supplied by 
the student. 

Closely connected with the foregoing theorem is the following : 
All spheres orthogonal to two fixed spheres form a bundle and all 
spheres orthogonal to three fixed spheres form a pencil. 

The foregoing assumes that the three spheres S^ S 2 , S s are 
not all planes. If they are, the bundle of spheres reduces to a 
bundle of planes. Otherwise the bundle of spheres contains a 
one-dimensional extent of planes through the radical axis of 
the bundle. 

113. Complexes of spheres. The spheres represented by the 
equation 8 l +v) t +p8.+ V 8<=0, (1) 

where S^ $ 2 , S 3 , S 4 do not belong to the same bundle or pencil 
and X, ft, v are arbitrary parameters, form a complex of spheres. 

The radical planes of the four spheres S^ $ 2 , $ 3 , $ 4 taken in 
pairs intersect in a point, and the radical plane of any two spheres 
of the complex pass through that point. This point is the radical 
center of the complex. From the properties of radical planes it 
follows that the square of the length of the tangents drawn from 
the radical center to all spheres of the complex is constant. There 
fore the radical center is the center of a sphere orthogonal to all 
the spheres of the complex. Conversely, it is easy to see that any 
sphere orthogonal to this sphere belongs to the complex. That is, 



270 THREE-DIMENSIONAL GEOMETRY 

the complex consists of spheres orthogonal to a fixed base sphere whose 
center is the radical center of the complex. 

If the four spheres intersect in a point that point is the radical 
center. The base sphere is then a sphere of radius zero and the 
complex consists of spheres passing through a point. 

The above discussion assumes that the four spheres S^ $ 2 , $ 3 , $ 4 
are not planes. If they are, the complex simply consists of all 
planes in space. In the general case the complex contains a doubly 
infinite set of planes which pass through the center of the base 
sphere. 

114. Inversion. Let be the center of a fixed sphere $, k 2 the 
square of its radius, and P any point. The point P may be trans 
formed into a point P by the condition that OPP forms a straight 
line and that OP-OP =k\ (1) 

This transformation is an inversion, or transformation by recip 
rocal radius. The point is the center of inversion, and the 
sphere S is the sphere with respect to which the inversion takes 
place. 

If the point has the coordinates (# , y o , g Q ), the equations of 
the transformation are 



i , 
0+ 






(2) 



where Sf= (x - x^ + (y - % ) 2 + ( Z - z o ) 2 . 

In this transformation the constants may be either real or 
imaginary. If (X Q , y Q , Z Q ) is real and k 2 real and positive, the 
inversion is with reference to a real sphere. If (X Q , y^ 2 o ) is real 
and k 2 real and negative, the inversion is with reference to a 
sphere with real center and pure imaginary radius. In this case, 
however, real points are transformed into real points. 

From the definition and equations (2) it appears that any point 
P has a unique transformed point P , and, conversely, unless P is 
at the origin, or on a minimum line through 0, or at infinity. 



THE SPHERE IN CARTESIAN COORDINATES 271 

To handle these special cases we take at the origin and write 
equations (2) with homogeneous coordinates as 



pz =k*zt, 

pt = X 2 +y*+z*. 

From (3) it appears that the transformed point of is indeter 
minate, but that if P approaches along the line x : y : z = I : m : n, 
the point P recedes to infinity and is transformed into the point at 
infinity I : m : n : 0. Hence we may say that the center of inver 
sion is transformed into the entire plane at infinity. Conversely, 
any point on the plane at infinity but not on the circle at infinity 
is transformed into 0. 

If P is on a minimum line through but not on the imaginary 
circle at infinity, then x f : y 1 : z x : y : z and t = 0. That is, all 
points on a minimum line through is transformed into the point 
in which that line meets the imaginary circle at infinity. Con 
versely, if P is on the imaginary circle at infinity the transformed 
point is indeterminate, but x : y : z f = x : y : z, so that any point on 
the circle at infinity is transformed into the minimum line through 
that point and the center of inversion. 

Consider now a sphere S with the equation 

a (> 2 + tf + z 2 ) + 2fx + 2 gy + 2 hz + c= 0. (4) 

It is transformed into 

ak* + Zftfx + 2 gtfy + 2 M 2 z + c(x* + y* + z 2 ) = 0. (5) 

This is in general a sphere, so that in general spheres are 
transformed into spheres. But exceptions are to be noted: 

1. If c = 0, a 3= 0, (4) is a sphere through and (5) a plane 
not through 0, so that spheres through the center of inversion are 
transformed into planes not through the center of inversion. 

2. If a = 0, c = 0, (4) is a plane not through and (5) a sphere 
through 0, so that planes not through the center of inversion are 
transformed into spheres through the center of inversion. 

3. If a = 0, c = 0, (4) and (5) represent the same plane through 0, 
so that planes through the center of inversion are transformed into 
themselves. 



272 THREE-DIMENSIONAL GEOMETRY 

By an inversion the angle between two curves is equal to the 
angle between the two transformed curves ; that is, the trans 
formation is conformal. To prove this we compute from (2) (with 



dx = {(*/ 2 + z 2 - x*) dx-2 xydy - 2 xzdz}, 

, (6) 



dz = <{- Zzxdx - 2 yzdy + <V+ y*- z^dz}. 

Hence, if we place ds 2 = dx * + dy *+ dz 2 and ds 2 = dx* + rf/ + dz\ 
we have p 



Now, if <fo, c??/, cfe correspond to displacements on a curve from 
P, and &r, 8y, 8z to displacements along another curve from P, the 
angle a between the curves is given by 

dx&x -h dy&y + dzz 

cosa= 



Similarly, the angle a 1 between the transformed curves is 

dx Sx +dy Sy +dz W 
- y ^ - 



and it is easy to prove from (6) that cos a = cos a . 

Any pencil, bundle, or complex of spheres is transformed into a 
pencil, bundle, or complex, respectively. The line of centers of the 
pencil is not, however, in general transformed into the line of cen 
ters of the transformed pencil, but becomes a circle cutting the 
spheres of the transformed pencil orthogonally. Also the radical 
plane of the pencil is not transformed into the radical plane of the 
transformed pencil, but into one of the spheres of that pencil. 

Similarly, the plane of centers of a bundle is transformed into a 
sphere cutting all the spheres of the bundle orthogonally, and the 
radical axis of the bundle is transformed into a circle orthogonal 
to the transformed bundle. 

On the other hand, the base sphere of a complex is transformed 
into the base sphere of the transformed complex. 



THE SPHERE IN CARTESIAN COORDINATES 273 

If we take a pencil of spheres intersecting in a real circle and 
take the center of inversion on that circle, the pencil of spheres is 
evidently transformed into a pencil of planes. If we take a bundle 
of spheres intersecting in two real points A and B, and take A as 
the center of inversion, the bundle of spheres becomes a bundle of 
planes through the inverse of B. If we take a complex of spheres 
and place the center of inversion on the base sphere, the complex 
becomes one with its base sphere a plane ; that is, it consists of all 
spheres whose centers are on a fixed plane. 

EXERCISES 

1. Prove that by an inversion with respect to a sphere S all spheres 
which pass through a point and its inverse are orthogonal to S. 

2. Prove that a point and its inverse are harmonic conjugates with 
respect to the points in which the line connecting the first two points 
intersects the sphere of inversion. 

3. Prove that the inverse of a circle is in general a circle and note 
the special cases. 

4. Prove that if two figures are inverse with respect to a sphere S 19 
their inverses with respect to a sphere S 2 whose center is not on 5 t are 
inverse with respect to S[, the inverse of S l with respect to 5 2 . 

5. Prove that if two figures are inverse with respect to a sphere S v their 
inverse with respect to a sphere S 2 whose center is on S 1 are symmetrical 
with respect to the plane P f , the inverse of S l with respect to S a . Con 
versely, if two figures are symmetrical with respect to a plane P they are 
inverse with respect to any sphere into which the plane P is inverted. 
Therefore inversion on a plane is defined as reflection on that plane. 

6. Prove that if S is a sphere of radius r and S is its inverse, the 
radius of S 1 is equal to the radius of S multiplied by the square of the 
radius of the sphere of inversion and divided by the absolute value of 
the power of the center of inversion with respect to S. 

7. Prove that any two nonintersecting spheres maybe inverted by 
an inversion on a real sphere into concentric spheres. 

8. Prove that any three spheres may be inverted into three spheres 
of equal radius. 

9. Prove that inversion on a sphere with real center and pure imagi 
nary radius ri is equivalent to inversion on a sphere with the same center 
and real radius r, followed by a transformation by which each point is 
replaced by its symmetrical point with respect to the center of inversion. 



274 THREE-DIMENSIONAL GEOMETRY 

10. A surface which is its own inverse is called anallagmatic. Prove 
that any anallagmatic surface cuts the sphere of inversion at right 
angles if the point of intersection is not a singular point of the surface 
and is the envelope of a family of spheres which cuts the sphere of 
inversion orthogonally. 

11. Prove that the product of two inversions is equivalent to the 
product of an inversion and a metrical transformation or in special cases 
to a metrical transformation alone. 

115. Dupin s cy elide. The transformation by inversion is useful 
in studying the class of surfaces known as Dupin s cyclides. These 
are denned as the envelope of a family of spheres which are tangent 
to three fixed spheres. 

If the centers of the fixed spheres do not lie in a straight line we 
may by inversion bring them into a straight line. To do this we 
have simply to draw, in the plane of the centers of the three 
spheres, a circle orthogonal to the three spheres and take any point 
on that circle as the center of inversion. The circle then goes into 
a straight line which is orthogonal to the three transformed spheres 
and hence passes through their centers. This is a consequence of 
the conformal nature of inversion. For the same reason the surface 
enveloped by spheres tangent to the original three spheres is in 
verted into a surface enveloped by spheres tangent to three spheres 
whose centers lie on a straight line. 

We shall study first the properties of such a surface and 
then by inversion deduce the properties of the general Dupin s 
cyclide. 

Let us take the line of centers of three fixed spheres as the axis 
of z and the equations of the spheres as 



Then, if the sphere 

6) 2 +( Z -<0 2 =>- 2 (2) 



is tangent to each of the spheres (1), the distance between the 
center of (2) and that of any one of the spheres (1) must be equal 



THE SPHERE IK CARTESIAN COORDINATES 275 

to the sum or the difference of the radii of the two spheres. This 
gives the three equations 

tf 2 + 2 +f 2 =(rr i y 2 , 

a 2 + 6 2 + <?- 2 c z c + <= (r r 2 ) 2 , (3) 



which have in general four solutions of the form 

c const., r = const., a~-\-b* = const. (4) 

Therefore the sphere (2) belongs to one of four families each 
of which consists of spheres with a constant radius and with 
their centers on a fixed circle. Each family obviously envelops a 
ring surface. 

There are therefore in general four Dupin s cyclides determined 
by the condition that the enveloping spheres are tangent to three 
fixed spheres. 

Let us take any one of the solutions (4) and change the coordi 
nate system so that c = 0. The equation of the family of spheres 
may then be written 

(x - a Q cos 0) 2 +(y-a Q sin 0) 2 + z 2 - r\ (5) 

where 6 is an arbitrary parameter and a Q and r are constants. 

The surface enveloped by (5) is 

<V + f + z *+ a 2_ r y = 4 <(:r + y> (6) 

This is the equation of the ring surface formed by revolving about 
the axis of z the circle , x _ a y _j_ ^ __ ^ ^ 

Hence any Dupin s cydide is the inverse of the ring surface formed 
by revolving a circle about an axis not in its plane. 

The ring surface contains two families of circles forming an 
orthogonal network. The one family consists of the meridian cir 
cles cut out by planes through the axis of revolution, the other of 
circles of latitude made by sections perpendicular to that axis. 

Since, by inversion, circles are transformed into circles, and angles 
are conserved, there exist on any Dupin s eyelid e two similar 
families of circles also forming an orthogonal network. 

The ring surface is the envelope not only of the family of spheres 
whose equation is (5) but also of the family with the equation 

(z - a tan 0) 2 = (a o sec - rf. (8) 



276 THREE-DIMENSIONAL GEOMETRY 

This family consists of spheres with their centers on OZ each of 
which may be generated by revolving about OZ a circle with its 
center on OZ and tangent to the circle (7). The spheres of this 
family are tangent to the ring surface along the circles of latitude, 
while the spheres of the family (5) are tangent to the ring surface 
along the meridian circles. The family of spheres (8) may be deter 
mined by the condition that they are tangent in a definite manner 
to three spheres of (5). 

Hence any Dupin s cy elide may be generated in two ways as the 
envelope of a family of spheres consisting of spheres tangent to three 
fixed spheres. Each family of spheres is tangent to the cyclide along 
a family of circles, the two families of circles being orthogonal. 

The planes of each family of circles intersect in a straight line. 
This follows from the theorems of 112, since the inverse spheres 
of the spheres (5) belong to the same bundle and the circles are inter 
sections of spheres of that bundle, so that their planes pass through 
the radical axis of the bundle. Similarly for the spheres (8). 

The circle (7) intersects the axis of z in two real, imaginary, or 
coincident points. Therefore a Dupin s cyclide has at least this 
number of singular points. .We shall see later that it also has 
other singular points, but we shall confine our attention at present 
to these two. Call them A and B. The spheres of one of the fami 
lies which envelop the cyclide intersect in A and B, as is seen in 
the case of the ring surface. Consequently, if one of these points, 
as A, is taken as the center of inversion this family of spheres 
becomes a family of planes, and the cyclide inverts into a surface 
enveloped by spheres which are tangent to three of these planes. 

If A and B are distinct the planes pass through the point B 1 , 
the inverse of B, and the cyclide is inverted into a cone of revolu 
tion, which is real if A and B are real, and imaginary if A and B 
are imaginary. 

If A coincides with B the planes are parallel and the cyclide is 
inverted into a cylinder of revolution. We have accordingly the* 
theorem: A Dupin s cyclide may always be inverted into a cone of revo 
lution which, in special cases, degenerates into a cylinder of revolution. 

Consequently we may obtain any cyclide in which the singular 
points A and B are distinct by inverting the cone 

2*+y-wV=0 (9) 



THE SPHERE IN CAETESIAN COORDINATES 277 

from any real or imaginary center of inversion with respect to any 
real or imaginary sphere ; or, what amounts to the same thing, we 
may transform the origin to any real or imaginary point and invert 
from the origin. The equation of the cone is then 

(x - *) 2 + 0/ - ) 2 - m\z - 7 ) 2 = 0, (10) 

and its inverse with respect to the origin is 

(c?+ 2 - wV) (x*+ y* + z 2 ) 2 - 2 k\ax + fry - 2 7 z) (x"+ y* + z 2 ) 

+ k\x 2 + y-wV) = 0. (11) 

To consider the case in which the points A and B coincide, we 

invert the cylinder 

( 9 -ay+(y-0)*=i* (12) 

and obtain for its inverse 



The cyclide is therefore a surface of the fourth order unless the 
first coefficient in either (11) or (12) vanishes. But this happens 
when and only when the cone (10) or the cylinder (12) passes 
through the center of inversion. 

If now we make the equations (11) arid (13) homogeneous, 
and place t = to determine the section with the plane at infinity, 
we get the circle at infinity as a double curve when the surface is 
of fourth order, and the circle at infinity, together with a straight 
line, when the surface is of the third order. 

Hence a Dupin s cyclide is a surface of the fourth order with 
the circle at infinity as a double curve, or a surface of the third order 
with the circle at infinity as a simple curve. 

We proceed to find the singular points of equation (11). We 
can without loss of generality so turn the axes that fi = Q, and 
will make the abbreviations 



L = ax m 2f yz, 
and write the equation as 

- 2 tfLR + k* (x 2 +y 2 - w 2 z 2 ) = 0. (14) 



278 THREE-DIMENSIONAL GEOMETRY 

The singular points are then the solutions of this equation and 
the following, formed by taking the partial derivatives with respect 
to x, #, and z : 

4 ARx - 2 tfaR - 4 tfLx + 2tfx = 0, 

4ARy -k*Ly + 2k*y = 0, (15) 

4 ARz + 2 kVyR - 4 tfLz - 2 k*m*z = 0. 



By multiplying equations (15) in order by z, y, z and adding, and 
subtracting the result from twice (14), we obtain 

(AR - T?L)R = 0. (16) 

Also, by combining the first two of (15) we have 

2 TfayR = 0. (17) 

From (17) we have either R = or y = 0. Taking first the 
condition y = 0, but R = 0, from (16) and (15), 

aR <yR 

k* 
whence R = - 



(ak 2 ryk 2 \ 

- -i 0, -~ - } is therefore a singular point. It is 
ar+7? or+<f/ 

the inverse of the vertex* of the cone and is the point B of the 
discussion on page 276. 

Consider now the solution R = of equation (17). From (15) 

& 2 
we have either x = 0, y = 0, z = 0, or L = > z = 0. The origin is 

2 

therefore a singular point, the inverse of the section of the cone 
with the plane at infinity, and is the point A of the discussion on 
page 276. 

The alternative R = Q,L= >z=0 leads to the two singular points 



case 



(k 2 k 2 i \ 
, ^i ). These points fail to exist if a 0, but in that 
2 a 2 a / 

the inversion is from a point on the axis of the cone, and the 
surface (11) is then a ring surface. 

The two singular points just found are each connected with A 
and B by minimum lines. 

If we consider in the same way equation (13), we obtain 
similar results except that the singular point B coincides with A at 



THE SPHERE IN CARTESIAN COORDINATES 279 

the origin, since the assumption y = leads to the conclusion R = 0. 

(k 2 k 2 i \ 
> J are again singular points unless a=0, 

when the surface (13) is a ring surface with a single singular point. 

A Dupin s cyclide which is not a ring surface has in general four 
finite singular points two of which are connected with the other two 
by minimum lines. Two of these singular points may coincide, in 
which case the cyclide has three finite singular points two of which 
are connected with the third by minimum lines. 

It follows, of course, that the Dupin s cy elides are not the gen 
eral surfaces of fourth order with the circle at infinity as a double 
curve nor the general surface of third order through the circle at 
infinity. These more general surfaces will be noticed in the next 
section. 

EXERCISES 

1. Prove that any Dupin s cyclide is anallagmatic with respect to 
each sphere of two pencils of spheres. 

2. Prove that the centers of each family of enveloping spheres of a 
Dupin s cyclide lie on a conic. 

3. Prove that the two lines in which the planes of the two families 
of circles on the Dupin s cyclide intersect are orthogonal. 

4. Prove that the circles on a Dupin s cyclide are lines of curvature. 
(A line of curvature on a surface is such that two normals to the surface 
at two consecutive points of the line of curvature intersect.) 

5. Prove that the only surfaces which have two families of circles 
for lines of curvature are Dupin s cyclides. (Exception should be made 
of the sphere, plane, and minimum developable, for which all lines are 
lines of curvature.) 

116. Cyclides. A cyclide is defined by the equation 

U Q (X*+ # 2 + z 2 ) 2 + u^ + y*+ z 2 ) + u = 0, (1) 

where U Q is a constant, u l a polynomial of the first degree, and w 2 a 
polynomial of the second degree in x, y, z. The Dupin s cyclides 
are special cases of the general cyclide. 

If U Q = in equation (1) the surface is of the fourth degree 
and represents a biquadratic surface with the imaginary circle at 
infinity as a double curve. 



280 THREE-DIMENSIONAL GEOMETRY 

If U Q = 0, equation (1) is a general of the third degree and repre 
sents a cubic surface passing through the imaginary circle at infinity. 

Degenerate cases of the cyclides may also occur if, in equation (1), 
u Q =0 and u^ is identically zero. The equation then represents a 
quadric surface or even a plane. These cases are important only 
as they arise by inversion from the general cases. 

In order to study the effect of inversion on the cyclide we may 
take the center of inversion at the origin, since the form of equation 
(1) is unaltered by transformation of coordinates. Such an inver 
sion produces an equation of the same form, which is of the fourth 
degree if u 2 contains an absolute term and of the third degree if u 2 
does not contain the absolute term but does contain linear terms. 
In the former case the origin is not on the surface ; in the latter 
case the origin is on the surface, but is not a singular point. Hence 

The inverse of any cyclide from a point not on it is always a cyclide 
of the fourth order. The inverse of any cyclide from a point on it 
which is not a singular point is always a cyclide of the third order. 

In general the cyclide will not have a singular point. If it does 
we may take it as the origin. Then in equation (1) the absolute 
term and the terms of first order in u 2 disappear. By inversion from 
the origin there will then be no terms of the fourth or the third 
degree. Hence the cyclide with a singular point is the inverse of a 
quadric surface. Conversely, as is easily seen, the inverse of a quadric 
surface is a cyclide with at least one singular point. 

Consider now a cyclide with two singular points A and B which 
do not lie on the same minimum line. If we invert from A the 
cyclide becomes a quadric surface with a singular point at B , the 
inverse of B. It is therefore a cone. Hence the cyclide with two 
singular points not on the same minimum line is the inverse of a quadric 
cone. Conversely, the inverse of a quadric cone from a point not on it 
is a cyclide with at least two singular points. 

We have shown in 115 that a Dupin s cyclide of the fourth 
order has in general four singular points. We shall now prove, 
conversely, that a cyclide of the fourth order with four singular 
points is a Dupin s cyclide. 

If the four points are A, B, C, D they cannot all be connected 
by minimum lines, since that is an impossible configuration. We 



THE SPHERE IN CARTESIAN COORDINATES 281 

will assume that A and B are not on a minimum line, and will 
invert from A, thus obtaining a quadric cone F with its vertex at 
J5 , the inverse of B. Any plane section of the cyclide through A B 
is a curve of the fourth order with two singular points at A and B 
and two other singular points on the circle at infinity. It therefore 
breaks up into two circles and is inverted into two straight-line 
generators of the cone F. The cone is enveloped by a one-parameter 
family of planes tangent along the generators. Therefore the 
cyclide is enveloped by a one-parameter family of spheres tangent 
along the circular sections through A and B. 

The plane section determined by the points J, B, and C has three 
singular points besides the two on the circle at infinity. Therefore 
it consists of a circle and two minimum lines, and since AB is not 
a minimum line, AC and BC are. By a similar argument AD and 
BD are minimum lines. Hence CD is not a minimum line. 

We may accordingly invert the cyclide from C and obtain another 
cone with the properties of F. In particular, the straight-line gen 
erators of this cone are the inverses of circles on the cyclide, and 
its tangent planes are the inverses of spheres tangent to the cyclide. 
Therefore the cone F is enveloped by spheres, the inverse with 
respect to A of the last-named family. Therefore F is a cone of 
revolution and, by 115, the theorem is proved. 

EXERCISES 

1. Prove that the envelope of spheres whose centers lie on a quadric 
surface and which are orthogonal to a given sphere is a cyclide. 

2. Discuss the plane curves called bicircular quartics, defined by the 

e<luatlon .(* +**>+ ,<<*+ >+-* 

and trace the analogies to the cyclides. 

3. Prove that the envelope of a circle which moves in a plane so that 
its center traces a fixed conic, while the circle is orthogonal to a fixed 
circle, is a bicircular quartic. 

4. The intersection of a sphere and a quadric surface is a sphero- 
quadric. Prove that a spheroquadric may be inverted into a bicircular 
quartic and conversely. 

5. Prove that the intersection of a cyclide and a sphere is a sphero 
quadric. 



CHAPTER XVI 

PENTASPHERICAL COORDINATES 

117. Specialized coordinates. Pentaspherical coordinates are based 
upon five spheres of reference, as the name implies. It is customary 
to define them by use of the Cartesian equations of the five spheres, 
but we prefer to build up the coordinate system independently of 
the Cartesian system, using only elementary ideas of measurement 
of real distance. This brings into emphasis the fact that penta- 
spherical coordinates are not dependent upon Cartesian coordinates, 
but that the two systems stand side by side, each on its own founda 
tion. One result is that certain ideal elements pertaining to the 
so-called imaginary circle at infinity which are found convenient in 
Cartesian geometry are nonexistent in pentaspherical geometry; 
and, conversely, certain ideal elements of pentaspherical geometry 
do not appear in Cartesian geometry. 

Let OX, OY, and OZ be three mutually perpendicular axes of 
reference intersecting at 0, P any real point, OP the distance from 
to P, and OL, OM, ON the three projections of OP on OX, OF, OZ 
respectively. Algebraic signs are to be attached to the three projec 
tions in the usual way, but OP is essentially positive. We may then 
take as coordinates of P the four ratios defined by the equations 

?, : f, : f, : f, : f.= ?* OL : OM: ON: 1 (1) 

and satisfying the fundamental relation * 

K+K+f 4 -f,f.= o. (2) 

It is obvious that to any real point corresponds a set of real 
coordinates and that to any set of real coordinates corresponds 
one real point. The extension to imaginary and infinite points is 
made in the usual manner. In particular, as P recedes from indefi 
nitely in any direction, the coordinates approach the limiting ratios 
1:0:0:0:0, which are the coordinates of a real point at infinity. 
This, however, is not the only point at infinity, as will appear when 
we consider the formula for the distance between two points. 

282 



PENTASPHERICAL COORDINATES 283 

The relation (1) may be reduced to a sum of squares by replacing 
the coordinates f . by new coordinates x { , where 



whence 



(4) 



and the coordinates x { satisfy the fundamental relation 



In these coordinates, which we shall use henceforth, a real point 
has four of its coordinates real and the fifth pure imaginary (the 
proportionality factor p being assumed real). This slight incon 
venience, if it is an inconvenience, is more than balanced by the 
symmetry of equation (5). The coordinates of the real point at 
infinity are now 1 : : : : i. 

If P l and P 2 are two real points with coordinates y { and x { respec 
tively, the projections of the line f^P 2 on OX, OY, OZ, respectively, 
are easily seen to be 




and hence, since the square of the distance of the line P^ is equal 
to the sum of the squares of its projections, we compute readily, with 
the aid of (5), the distance formula for the distance d between two 

P intS 2 x x x + x& + x,y^ ? g 



which is the same as 

CO 



a*(x, y) being the polar of o>(V). 



284 THREE-DIMENSIONAL GEOMETRY 

The formula (6), thus derived for real points, will be taken as 
the definition of distance between all kinds of points. From this it 
appears that d is infinite when and only when one of the points 
satisfies the equations 2^+ ix^= and o>(#, y)^ 0. Hence the locus 
of points at infinity is given by the equation x^+ix^ = 0. 

Since the coordinates of all points satisfy (5), we have for points 
at infinity x 1 + ix 6 = and x% + xl -f xl = 0. Therefore the point 
1 : : : : i is the only real point at infinity. The nature of the 
imaginary locus at infinity will appear later. 

118. The sphere. A sphere is defined as usual as the locus of 
points equally distant from a fixed point. This definition includes 
all spheres in the usual sense and all loci which are expressed by 
equation (6), 117, in which y i is fixed and d = r a constant. This 
equation is 



+[2 ?.+ <>!+ &)*]*.= o. (i) 

This is of the type 



where pa l = 2 ^ + (^ + iy^) r\ 

P a 2 =2y 2 > 

/>V=2y., (3) 

P a i=2ys 

p a ,= % y 5 + *(*/!+ iy^r* 

From these equations and the fundamental relation o> (?/) = 0, 




. a 



which give the center and the radius of any sphere (2) in terms of 
the coefficients a { . We have, then, the following statement, half 
theorem, half definition. 



PENTASPHERICAL COORDINATES 285 

Every linear equation of the type (2) represents a sphere, the center 
and the radius of which are given by equations (4). 

It is convenient to represent by 77 (a) the numerator of r 2 in (4) ; 

that is, ^\_2,2,2j_2_,_2 

t] \ > ) **i ~t~ ^2 ^* 3 "i ^4 i a^ . 
We have, then, the following classes of spheres : 
CASE I. 77 (a) = 0. Nonspecial spheres. 

Subcase 1. 77 (a) = 0, a^+ia^ 0. Proper spheres. The center and 
the radius of the sphere is finite, but neither is necessarily real. 
The sphere does not contain the real point at infinity. 

Subcase 2. 77 (a)^ 0, ^-f ia & = 0. Ordinary planes. The radius 
is infinite. The center is the real point at infinity. Since a plane 
is the limit of a sphere with center receding to infinity and radius 
increasing without limit, we shall call this locus a plane. This 
may be justified by returning to the coordinates f t .. The equa 
tion then reduces to 2 f 2 4- a 8 Z 3 + a c~ a ^~ w ^ ^ e condition 
# 2 2 + # 2 4- a l^ 0- By repetition of the familiar argument of analyti 
cal geometry this may be shown to represent a plane. 

Since this case differs from the previous one essentially in that 
the coordinates 1 : : : : i now satisfy the equation of the sphere, 
we may say : A proper plane may be defined as a nonspecial sphere 
which passes through the real point at infinity. 

CASE II. 77 (a) = 0. Special spheres. 

Subcase 1. 77 (a) = 0, d^-fia^O. Point spheres. The radius is 
zero and the center is not at infinity. It is obvious that the sphere 
passes through its center y. = a., and if y i is real the sphere con 
tains no other real point. The sphere does not contain the real 
point at infinity. 

Subcase 2. 77 (a) = 0, a l -\-ia 5 = 0. Special planes. The radius is 
indeterminate. The center is ajaj a 3 : a^: ia^ which is a point at 
infinity. The equation of the sphere may be written 

which, in Cartesian geometry, would be that of a minimum plane 
(80). In this case the sphere contains the real point at infinity. 
Hence we may say: A special plane is a point sphere which 
passes through the real point at infinity. 



286 THREE-DIMENSIONAL GEOMETRY 

The locus at infinity is, as we have seen, x^+ ix & = 0. This comes 
under Case II, Subcase 2, and is therefore a special plane with its 
center at 1 : : : : i ; that is, the locus at infinity is a special 
plane whose center is the real point at infinity. 

119. Angle between spheres. The angle between two real proper 
spheres is equal or supplementary to the angle between their radii 
at any point of intersection. For precision we will take as the 
angle that one which is in the triangle formed by the radii to the 
point of intersection and the line of centers of the spheres. If 6 
is this angle, d the distance between the centers, and r and r 1 the 
radii, then 



, 2 2 . 2 . a 
er= r+ r * 2 rr cos 6. 

If now the equations of the two spheres are 

2^ = 0, 2)** = 

an easy calculation by aid of formulas (4), 118, and (6), 117, 

( ^ { r , a . 



whence 

COS e = , aib ^ ^ 2 2+ ^f 8 ) JT 4 a "^ 5 2 a a * C 1 ) 

This formula has been derived for real proper spheres intersect 
ing in real points. We take it as the definition of the angle 
between any two spheres. The student may show that if one or 
both of the two spheres becomes a real plane, this definition of 
angle agrees with the usual one. 

Two spheres ^a f x t = 0, ^.bjK i = are 



If both of the spheres are nonspecial, this agrees with the usual 
definition. If, however, ^a f x t = is a special sphere, the condi 
tion expresses the fact that the center of ^^f = ^ ^ es on ^ e 
sphere ^b { x { = 0. Hence 

The necessary and sufficient condition that a special sphere should 
be orthogonal to another sphere is that the center of the special sphere 
lie on the other sphere. 



PENTASPHERICAL COORDINATES 287 

EXERCISE 

Prove that the coefficients a { in the equation of the sphere are pro 
portional to the cosines of the angles made by the sphere with the 
coordinate spheres, and that the cosines themselves may be found by 
dividing a t by V^ 2 + a 2 2 + 8 2 + a? + 5 2 . Compare with direction cosines 
in Cartesian geometry. 

120. The power of a point with respect to a sphere. If C is the 

center of the sphere 



with the radius r, and P is any point with coordinates y^ the dis 
tance CP is easily calculated by (4), 118, and (6), 117, with 
the result: 



(!+. 

We shall place 

2 s 2 Q^ -f a 



, 2 

and shall call ^ the power of the point y { with respect to the sphere. 
If the sphere is real and the point y i is a real point outside the sphere, 
the power is the square of the length of any tangent from the point 
to the sphere. If the sphere is a point sphere, the power is the square 
of the distance from the point y { to the center of the sphere. In all 
other cases equation (2) is the definition of the power. 

From (2) may be obtained the important formula for a non- 
special sphere : 

S = 2 a 1 y 1 + a^ + a 3 */ 3 + ^4 + <W* . ^ 

r 1/i+iff* V! 2 + 2 2 +3 2 +<+ 5 2 

The above discussion fails if the sphere is a plane. We may, 
however, obtain the meaning of formula (3) in this case by a limit 
process. We have, from (2), 

S = (P C - r) (P C + r) = PA (P C + r) , 
where PA is the shortest distance from P to the sphere. Then 



288 THREE-DIMENSIONAL GEOMETRY 

Now let C recede to infinity along the line PC. The sphere 

PC 
becomes a plane perpendicular to PA. But the limit of - as 

r becomes infinite and a^ ia & approaches zero, is 1, from (1). 
Therefore 



where PA is the perpendicular from P to the plane. This result 
may be checked by replacing x i by and using familiar theorems 
of Cartesian geometry. 

The equation of any nonspecial sphere may be written so that 
17 (a) =1. The equation is then said to be in its normal form, and 
the denominator a?+ a% + a% + a* + 0% disappears from equation (3). 

121. General orthogonal coordinates. Let us make the linear 
substitution 

pat = a^x, + a i2 x 2 + a is x s + a rt z 4 + a i5 x 6 , (i = 1, 2, 3, 4, 5) (1) 

in which the determinant a ik \ does not vanish. Then to any set of 
ratios x i corresponds one set of ratios zj, and since the quantities x i 
satisfy a quadratic relation &> (x) = 0, the quantities x( satisfy another 
quadratic relation ft (V) = 0. 

Then values of x( which satisfy fl (V) = correspond to one and 
only one set of ratios of x i which satisfy o> (x) = 0. Therefore x[ 
can be taken as coordinates of a point in space and are the most 
general pentaspherical coordinates. 

The sphere 2X^,= ^ 

becomes the sphere ^ 

where pa, = a u a( + a. 2i a 2 

and the condition 77 (a) = for a special sphere goes into another 
quadratic condition H (V) = 0. 

The point at infinity takes the new coordinates a fl + ia^ , and the 
condition that a sphere should be a plane is that its equation should 
be satisfied by these coordinates. 

The coordinates of 117 form a special case of these general 
coordinates. We shall not, however, pursue the treatment of the 
general case, but shall restrict ourselves to the case in which the 
five coordinate spheres are orthogonal. In this case no sphere can 
be special, since, if it were, its center would lie on each of the other 



PENTASPHERICAL COORDINATES 289 

four spheres, and there would be four orthogonal spheres through 
a common point, which is obviously absurd. 

We may consider that each of the equations of the coordinate 

spheres has been put in the normal form, so that we have, in (1), 

?,+ +<+<+<=!. (3) 

Then, by (3), 120, the substitution is expressed by the equations 

&=*> (4) 

where S t is the power of the point x i with respect to the sphere 

9 

x . = 0, and r, is the radius of x . = 0, since the factor -- - is the 

x, + ix 5 

same for all five spheres. If any sphere x k = is a plane, then 

Cf 

the corresponding term is to be replaced by 2P kJ where P k is 

r k 

the length of the perpendicular from x i to the plane x f k = 0. 
Since the five spheres in (3) are orthogonal we have 



for all pairs of values of i and k, i = k. 

From a familiar theorem of algebra on orthogonal substitutions * 
it follows that < + <+<.+ < + < = ! -(6) 



Consequently we have for x\ the fundamental relation 

*; 2 + 2 f+*f + 3 i 2 + a f=o, (8) 

and the condition for a special sphere is 



i a a v x 

Moreover, by the theory of orthogonal substitutions, equations (1) 
solve into ^ = /o(v ; + a ^ + a ^ + a ti x( + a^nf). (10) 

By (4), 118, the radius r . of the sphere x[= is 

r^^^ (11) 

Therefore the real point at infinity whose coordinates in the old 
system x i are 1 : : : : i* has the new coordinates 

/>*;=- c 12 ) 

r i 

* Cf. Scott s "Theory of Determinants," p. 154. . 



290 THREE-DIMENSIONAL GEOMETRY 

where, if any sphere x k = is a plane, the corresponding coordinate 
x k is zero, as in fact happens when r k oo. 

The equation x^ ix & = for the locus at infinity becomes, 
from (10) and (11), r t 

2^ = , ( 13 > 

where, again, if any coordinate sphere is a plane the corresponding 
term vanishes from (13). 

It is now easy to see that the formula (6), 117, for distance 
becomes ^^ 2 (^+^+ xjy B + aX+*iyi) , (14) 



so that the equation of a sphere with center y { and radius r is 

22>X-+^*Lo. (15) 

Identifying this with ]a[zj = (16) 

we have X= yS+ - (17) 



From (11), with (3) and (5), 



so that, from (17), p^-=^-. (19) 

By squaring (17), adding, and reducing by (8), (18), and 
(19), we obtain the following formulas for the radius and the 
center of the sphere (16): ^ 



(20) 



The formulas of 118 are only special cases of these. 

EXERCISES 

1. Prove the relation V-* = 2. 

2. Deduce for the element of arc ds z = 



($f 



PENTASPHERICAL COORDINATES 291 

122. The linear transformation. Consider a linear transformation 



in which the determinant \a ik \ does not vanish and by which the 
fundamental relation w(x) = is invariant. Then the relation 
77(0) = is also invariant. 

The relations (1) define a one-to-one transformation of space 
by which a nonspecial sphere goes into a nonspecial sphere and a 
special sphere into a special sphere. There are two types to be 
distinguished. 

I. Transformations by ivhich the real point at infinity is invariant. 
By such a transformation planes are transformed into planes and, 
consequently, straight lines into straight lines. Since the trans 
formation is analytic it is a collineation. 

Point spheres are transformed into point spheres ; therefore, 
expressed in Cartesian coordinates, the transformation is one by 
which minimum cones go into minimum cones, and consequently 
the circle at infinity is invariant. Hence the transformation is a 
metrical transformation. 

Conversely, any metrical transformation may be expressed as a 
linear transformation of pentaspherical coordinates. This is easily 
seen by use of the special coordinates of 117 and is consequently 
true for the general coordinates. 

Hence a linear transformation of pentaspherical coordinates by 
which the real point at infinity is invariant is a metrical transformation, 
and conversely. 

II. Transformations by which the real point at infinity is not inva 
riant. Among these transformations are the inversions. That an 
inversion may be represented actually by a linear transformation 
of pentaspherical coordinates is evident from the example in the 
coordinates &, 117, = & 4 



and in fact any inversion may be so expressed by proper choice 
of coordinates. 



292 THREE-DIMENSIONAL GEOMETRY 

Consider now the general case of a real transformation by which 
the real point at infinity / is transformed into a real point A, and 
the same point A, or another point A , is transformed into I. 
Since the transformation is real A cannot be at infinity. Let this 
transformation be T and let S be an inversion with A as the center 
of inversion. Then the product ST leaves / invariant and is there 
fore a metrical transformation, M. Therefore ST=M; whence 
T = S- 1 M. B ut S~ l = S. Therefore T = SM. Hence 

Any real transformation of pentaspherical coordinates by which the 
real point at infinity is not invariant is either an inversion, or the 
product of an inversion and a metrical transformation. 

This does not exhaust all cases of imaginary transformations. 
We may obviously have imaginary transformations of the metrical 
type or inversions from imaginary points, so that the above theorems 
hold for transformations by which the real point at infinity is trans 
formed into itself or into any finite point. Transformations, however, 
by which the real point at infinity is transformed into an imaginary 
point at infinity are of a different type. An example of such a 

transformation is ,_ . 

px l x 1 A x s ^^ 5 , 






We shall close this section with the theorem, important in subse 
quent work : If the coordinate system is orthogonal the transforma 
tion expressed by changing the sign of one of the coordinates is an 
inversion on the corresponding coordinate sphere. 

For let the sign of x k be changed. Then points on the sphere 
x k = are unchanged, and any sphere orthogonal to x k = is trans 
formed into itself. This characterizes an inversion on x k = 0. 

EXERCISES 

1. Prove the last theorem analytically, using the formulas of 121. 

2. Prove that the product of five inversions with respect to five 
orthogonal spheres is an identity. 



PENTASPHERICAL COORDINATES 293 

123. Relation between pentaspherical and Cartesian coordinates. 
If we take the axes OX, OY, OZused in 117 to define the special 
ized pentaspherical coordinates as the axes also of a set of Cartesian 
coordinates, it is obvious that we have, for real points, 



=20, (1) 

= 2zt, 



This establishes in the first place a one-to-one correspondence 
between real points in the two systems. It may be used also to 
define the correspondence between the imaginary and infinite points 
introduced into each system. There exists, however, no reason 
why such points introduced into one system should always have 
corresponding points in the other. As a matter of fact a failure of 
correspondence of such points does exist. 

The Cartesian points on the imaginary circle at infinity fail to exist 
in pentaspherical coordinates since values of #, y, z, t which satisfy the 
relations x 2 + y* + z* = 0, t = give ^ : x 2 : x 3 : x 4 : x^ = : : : : 0. 
But any Cartesian point at infinity not on the imaginary circle 
corresponds in pentaspherical coordinates to the real point at 
infinity 1 : : : : i. 

On the other hand, we have in pentaspherical geometry imaginary 
points at infinity satisfying the relations x% + xl + xl = 0, x l -f ix b = 0, 
but not having # 2 = x s = x 4 = 0. These have no corresponding points 
in Cartesian geometry since no values of x : y : z : t in (1) give them. 

This failure in the correspondence is of importance if one wishes 
to pass from one system to the other. They are of no significance, 
however, as long as one operates exclusively in one system. 

The general pentaspherical coordinates are connected with Car 
tesian coordinates by equations of the form 

px( = (a tl + ia.. ) (z 2 + / + z 2 ) + 2 a a x + 2a.,y+2 a,, z - (a a - ia ib ). 



124. Pencils, bundles, and complexes of spheres. If a.^.= and 
5^= are two spheres, the equation 

+ Xi,.> ; =0 (1) 



294 THREE-DIMENSIONAL GEOMETRY 

represents a sphere through all points common to the two spheres 
and intersecting neither in any other point. Such spheres together 
form a pencil of spheres. 

A pencil of spheres contains one and only one plane unless it con 
sists entirely of planes. 

This follows from the fact that the condition that equation (1) 
should be satisfied by the coordinates of the real point at infinity 
consists of an equation of the first degree in X, unless both 
Va i a; l .= and 2^~ ^ are satisfied by those coordinates. In the 
latter case all the spheres (1) are planes. 

A pencil of spheres contains two and only two special spheres (which 
may be real, imaginary, or coincident) unless it consists entirely of 
special spheres. 

The condition that (1) represents a special sphere is 
i/(a + X6) = i7() + Xiy(a, b) + \ 2 rj (6) = 0, 

which determines two distinct or equal values of X unless rj (a) = 0, 
77 (5) = 0, 77 (a, 6) = 0. The latter case occurs when the two spheres 
2^2-; = 0, V &,-#,- = are special spheres with the center of each on 
the other. 

The theorems of 111 and others analogous to those of 62 are 
easily proved by the student. 

If 2}^= 2)^i= ^ c i x iQ are three spheres not in the 
same pencil, the equation 



represents a bundle of spheres as in 112. The bundle contains 
a singly infinite set of planes and a singly infinite set of special 
spheres. The relations between orthogonal pencils and bundles 
found in 112 are easily verified here. 

If 2^ t =0, 5)^=0, 2^=0, 2/ta= are four spheres 
not belonging to the same bundle, the equation 

= 



represents a complex of spheres. It consists of spheres orthogonal 
to a base sphere and contains a doubly infinite set of planes and a 
doubly infinite set of special spheres. The centers of the latter 
form the base sphere. 



PENTASPHERICAL COORDINATES 295 

EXERCISES 

1. Prove that the angle under which a sphere cuts any sphere of a 
pencil is determined by the angle under which it cuts two spheres of 
the pencil. 

2. Prove that among the spheres of a pencil there is always one 
which cuts a given sphere orthogonally. 

3. Prove that the angle under which a sphere cuts any sphere of a 
bundle is determined by the angles under which it cuts three spheres 
of the bundle. 

4. Determine a sphere orthogonal to four given spheres. 

5. Determine a sphere cutting five given spheres under given angles. 
When is the problem indeterminate ? 

125. Tangent circles and spheres. Let y t , z t ., t t be any three 
points given in orthogonal pentaspherical coordinates, and consider 
the equations px _ = y< + x^+ ^.. (1) 

In order that x i should be the coordinates of a point it is neces 
sary and sufficient that 

2j(y*+H+^) =o. (2) 

Since ^y\= 0, ^zf= 0, *% = 0, equation (2) reduces to 

A\ + BIL + C\fjL = 0, (3) 

where A = ^yfr, B = ^yf^ C= ^^. 
Therefore (1) may be written 



or px i = By i + (Cy. + Bz i - At^) \ + Cfe.X 2 . (5) 

This represents a one-dimensional extent of points. Any sphere 
which contains the three points y^ z it t t will also contain all the 
points x it and any point x i belongs to all the spheres through ?/., t ., t .. 
Therefore (4) represents a circle, including the special case of a 
straight line. 

Any equation f(x l9 # 2 , 8 , x^ x 6 ) = 0, (6) 

where / is a homogeneous polynomial of the nth degree, represents 
a surface. To find where it is cut by any circle substitute from 
(5) into (6). There results an equation of degree 2 n in X, so that 
the surface is cut by any circle in 2 n points. 



296 THREE-DIMENSIONAL GEOMETRY 

If Cartesian coordinates are substituted for x i in (6) the equation 
is of the 2wth order and of the form 



where u k is a homogeneous polynomial of degree k not containing 
^ 2 _l_^ 2 _l_ 2 2 ) as a factor. The surface therefore contains the circle 
at infinity and as an w-fold curve if U Q = 0. In the Cartesian 
geometry the surface is cut by any circle in 4 n points, but the cir 
cular points at infinity count 2n times and do not appear in the 
tetracyclical geometry. 
The equation in X is 

A^ + = 0. (7) 



Now if y t is on the surface, then /(/) = and ^y. -j- = 0, the 

7t 

latter because /is homogeneous. Therefore one root of (7) is zero. 
Two roots will be zero if, in addition to y i being on the surface, 
we have , 

*s?*-^i*-* 

**fy. **%, 

which is the same as 

^* - 

(8) 



If this condition is satisfied by the two points z i and f t ., the circle 
(1) is tangent to the surface (6) at y { . The condition is certainly 
met if z i and t i are both on the same sphere of the pencil 



Any sphere of this pencil has accordingly the property that any 
plane section of it through y i is a circle tangent to the surface (6). 
Therefore (9) represents a pencil of tangent spheres to the surface. 

If -*-= 0, all circles through y t meet the surface in two coinci- 

U i 

dent points. The point y. is therefore a singular point. It is 
obvious that the geometric meaning is the same as in the Cartesian 
geometry. 



PENTASPHERICAL COORDINATES 297 

126. Cyclides in pentaspherical coordinates. Consider the surface 

]^Vz* = 0. (=*) (1) 

From 123 and 116 this is a cy elide. We have shown that if the 
cyclide has singular points, it is the inverse of a quadric surface. 
We shall therefore limit ourselves here to the general case in which 

the singular points do not exist. Since, then, the equations = 

tyi 
have no common solution, it is necessary and sufficient that the 

discriminant | a ik does not vanish. 

It is a theorem of algebra that in this case the quadratic form 
may be reduced by a linear substitution to the form 

0^+^x1+ c z xl+cixt+c b x%=^ (2) 

(where c^ 0), at the same time that the fundamental relation 
"^ iS ^+2^+ ^+^+^=0. (3) 

We shall therefore assume that the equation of the cyclide is in 
the form (2) and that the coordinates are orthogonal. 

From equation (2) it is obvious that the equation of the surface 
is not altered by changing the sign of any one of the coordinates ar t , 
But this operation is equivalent to inversion on the sphere x t = 0. 
Hence 

The general cyclide is its own inverse with respect to each of five 
mutually orthogonal spheres. 

The pencil of tangent spheres to the cyclide at any point y i is, 
by 125, 



+*)^-=0. (4) 

Hence, in order that a given sphere 



should be tangent to (2), it is necessary and sufficient to determine 
\ and y. so that 



and so that y. should satisfy the three equations (2), (3), (5). 
This gives the three conditions 



( +V> 



298 THREE-DIMENSIONAL GEOMETRY 

of which the first is a consequence of the last two. The last two 
express the fact that the equation 

= (8) 



*i+X 

has equal roots. This imposes a condition to be satisfied in order 
that (5) should be tangent to (2). 

When X has been determined from these equations, equations (6) 
determine y i in general without ambiguity. Exceptions occur if 
\ = c k , where c k is any one of the coefficients of (2). In that case 
we have in (6) a k = 0, and y k cannot be determined from (6). How 
ever, if the other four coordinates y i are determined, y k has two 
values of opposite sign but equal absolute value, determined from 
the fundamental relation (3). The corresponding sphere (5) is 
orthogonal to x k = and tangent to the cyclide at two points which 
are inverse with respect to x k 0. 

The value of X may be taken arbitrarily as c k ; whence a k = 0. 
The values of ( .(i=#) must then be determined from (7) with 
X = c k . Each of the first two equations contain an indetermi 
nate term. The last equation becomes 

Sr^-r = - (* **) (9) 



The coefficients of (5) satisfy two equations, therefore, and the 
spheres form a family of spheres which is not linear. In this family 
a sphere can be found which is tangent to the cyclide at any 
given point. For if X = c t , and y i is any point on the cyclide, 
equation (6) will determine a,., and the a/s will satisfy (9), as 
has been shown. The spheres of the family therefore envelop 
the cyclide. 

There are five such families of spheres, since X may be any one 
of the five coefficients <?,.. Hence 

The general cyclide is enveloped by jive families of spheres, each 
family consisting of spheres orthogonal to one of the five coordinate 
spheres and tangent to the surface at two points. 

We shall show that the centers of the spheres of each series lie on 
a quadric surface. 



PENTASPHERICAL COORDINATES 299 

Take, for example, the series for which \ = c l and a^ = 0. If 
y i are the coordinates of the center of a sphere of the series by (20), 



and 



whence pa k = 

and equation (9) becomes 



= 2,3,4,5) (10) 



which is the equation of the locus of the centers of the spheres of 
the family under consideration. 

By (4), 121, equation (10) may be written 



and, finally, if S t and ^ are expressed in Cartesian coordinates, 
equation (11) is of the second degree, and the theorem is proved. 
We may sum up in the following theorem : 

The general cy elide may be generated in five ways as the envelope of 
a sphere subject to the two conditions that it should be orthogonal to a 
fixed sphere and that its center should lie on a quadric surface. 

A surface which is its own inverse with respect to a sphere S 
is called anallagmatic with respect to , which is called the direc 
trix sphere. Such a surface is enveloped by a family of spheres 
orthogonal to S and doubly tangent to the surface. For at any 
point P of the surface there is a sphere tangent to the surface and 
orthogonal to S. By inversion this sphere is unchanged. It is 
therefore tangent to the surface at P , the inverse of P. 

The surface on which the centers of these enveloping spheres 
of the anallagmatic surface lie is called the deferent. 

The cyclide, therefore, is anallagmatic with respect to the five orthog 
onal spheres and has five deferents, each a quadric surface. 



300 THKEE-DIMENSIONAL GEOMETRY 

EXERCISES 

1. If Q t is one of the five deferents of the cyclide, and S t the corre 
sponding directrix sphere, prove that the tetrahedron whose vertices 
are the centers of the other five directrices is self-conjugate, both with 
respect to Q k and with respect to S k . 

2. Prove that on the cyclide there are ten families of circles, two 
families corresponding to each of the five modes of generating the 
cyclide. 

3. The focal curve of any surface being defined as the locus of the 
centers of point spheres which are doubly tangent to the surface, prove 
that the cyclide has five focal curves, each being a sphero-quadric formed 
by the intersection of a deferent by the corresponding directrix sphere. 

REFERENCES 

For more reading along the lines of Part III of this book the following 
references are given. As in Part II, these are not intended to form a complete 
bibliography or to contain journal references. 

General treatises 

CLEBSCH-LINDEMAN, Vorlesungen iiber Geometrie. Teubner. 
DARBOUX. See reference at end of Part II. 
NIEWENGLOWSKI, Ge ome trie dans 1 espace. Gauthier-Villars. 
SALMON-ROGERS, Geometry of Three Dimensions. Longmans, Green & Co. 

Circle and spheres : 

COOLIDGE, Circle and Sphere Geometry. Oxford Clarendon Press. 

Cydides : 

BOCHER, Potentialtheorie. Teubner. 

DARBOUX, Sur une classe remarquable de courbes et de surfaces alge"briques. 

A. Hermann. 
DOEHLEMANN, Geomctrische Transformationen, II. Teil. Goschen sche Verlags- 

handlung. 

These are in addition to the general treatises of Darboux and Coolidge already 
referred to. The first section of Bocher s book is of interest here. Doehlemann 
contains figures of the Dupin cyclides. 

Algebraic operations: 

BOCHER, Introduction to Higher Algebra. The Macmillan Company. 

BROMWICH, Quadratic Forms and their Classification by Means of Invariant 
Factors. Cambridge University Press. 

Each of these books contains geometrical illustrations. The student may refer 
to them for any algebraic methods we have employed and especially for an expla 
nation of the method of elementary divisors in reducing one or a pair of quadratic 
forms to various types. Bromwich contains a full classification of the cyclides. 



PART IV. GEOMETRY OF FOUR AND HIGHER 
DIMENSIONS 

CHAPTER XVII 

LINE COORDINATES IN THREE-DIMENSIONAL SPACE 

127. The Pliicker coordinates. The straight lines in space form a 
simple example of a four-dimensional extent, since a line is deter 
mined by four coordinates. In fact, the equations of a line can 
in general be put in the form 

x = rz + p, _ 

y = sz + (T, 

and the quantities (r, s, />, cr) may be taken as the coordinates of 
the line. More symmetry is obtained, however, by the following 
device. 

From equations (1) we have 

ry-sx = r(T-ps, (2) 

and we may place ra- ps = 77, (3) 

thus obtaining five coordinates connected by a quadratic relation. 

If (V, y, z ) and (x", y", z"} are any points on the line (1), we 
may easily compute 
r:s:p:v:rj:l = x -x f :y -y":x z -x z":y"z f -y f z":x f y"-x"y :z -z", 

and it is the ratios on the right-hand side of this equation which 
were taken by Pliicker as the coordinates of a line. 

These coordinates, however, form only a special case, arising 
from the use of Cartesian coordinates, of more general coordinates 
obtained by the use of quadriplanar coordinates. We proceed to 
obtain these coordinates independently of the work just done. 

The position of a straight line is fixed by two points (x^.x^x^x^) 
and (jjjyjyjy^)- It should be possible, therefore, to take as coor 
dinates of the line some functions of the coordinates of these two 
points. Furthermore, since any two points whose coordinates are 

301 



302 



FOUR-DIMENSIONAL GEOMETRY 



\x { + ^y { may be used to define the same line as is denned by x { 
and y^ the coordinates of the line must be invariant with respect 
to the substitutions 

px( = X^ t . + n$ py i = \Xi + pj/r 
Simple expressions fulfilling these conditions are the ratios of 



. We will, accordingly, consider 



determinants of the form 
the expressions 



Since ^.. = p. k , there are six of these quantities; namely, 



which are connected by the relation 
x i 

y\ 

x i 
y 

It is obvious that to any straight line corresponds one and only 
one set of ratios of the quantities p ik . 

As we have seen, the ratios of p ik are independent of the partic 
ular points of the line used to form p ik . If in particular we take 
one point as the point : x z : x z : x^ in which the line cuts the plane 
x t = 0, we have P 12 =^^ P ls = X 3 ^^ P 14 =~ x 4 y l l whence 
x z :x z : x t = P\2 : Piz : Pu Using in a similar manner the points in 
which the line meets the other coordinate planes, we have, as the 
points of intersection with the four planes, the following four points : 





--Pi. : - 







LINE COORDINATES 303 

The condition that these four points should lie on a straight 
line is exactly the relation (4). 

From (5) it follows that a set of ratios p ik can belong to only 
one line and that these ratios may have any value consistent 
with (4). 

Hence the ratios of p ik may be taken as the coordinates of a straight 
line, and the relation between a straight line and its coordinates is one 
to one. These coordinates are called Pl dcker coordinates. 

Of course if a straight line lies completely in one of the coor 
dinate planes, one of the sets of ratios in (5) becomes indeterminate. 
This cannot happen, however, for more than two of the sets at the 
same time, and the other two sets, together with (4), determine^. 

128. Dualistic definition. A straight line may be defined by the 
intersection of two planes u. and v { . Reasoning as in 127 we are 
led to place 



which are connected by the relation 



To any straight line corresponds one ratio set of ratios of q ik , 
and the four planes through the straight line and the vertices of 
the tetrahedron of reference have the plane coordinates 

: U : q u : q u , 

-q : : &.:-?, 

-ll -fa : : t (3) 

-<? 9 : -? : - 

Therefore, to any set of values of the six quantities q lk which 
satisfy the relation (2), there corresponds one and only one line 
with the coordinates q ik . 

The relation between the quantities p ik and q ik is simple. From 
(3) the plane 



304 



FOUR-DIMENSIONAL GEOMETRY 



passes through the line q ik . If x i and y k are two points on the line 
we have, besides equation (4), the equation 

+ = 0. 



From (4) and (5) we have 



3 = ii-s = 2". 

^34 ^42 # 



Similarly, we may show that 



_ i4 5*2 



We may, accordingly, use only one set of quantities : 



bound by the fundamental relation 

(r) = 2 (r 12 r 34 + r 13 r 42 + r 14 r 23 ) = 0, 

and may interpret in point or plane coordinates at pleasure. 

129. Intersecting lines. Two straight lines, one determined by 
the points x { and y. and the other by the points x( and y t , inter 
sect when the four points lie in the same plane, and only then. 
The necessary and sufficient condition for this is 



-o, 



which is the same as 



(1) 

Also, dualistically, two lines, one determined by the planes u { 
and v { and the other by the planes u( and v[., intersect when the 



LINE COORDINATES 305 

four planes pass through the same point, and only then. The 
necessary and sufficient condition for this is 
u, w, w, U A 



= 0, 

Mi K z M, U\ 

v[ v 2 v 8 v( 
which is the same as 



1 M 2 "I ^4 

v l v 2 v s v 
u( v( J u t 



Either condition (1) or (2) is in terms of r ik , 

**/**+ r i/u+ *V44 r 3 X 2 4- Vis 4- Vu = 0, (3) 

which is more compactly written as 



where &>(>, /) is the polar of the quadratic expression a>(r). 

130. General line coordinates. Consider any six quantities x i de 
nned as linear combinations of the six quantities r ik . That is, let 

P*i = Via 4 Vis + a is r i4 4- a^ M + a i5 r 42 + a^g, 
with the condition that the determinant of the coefficients 
does not vanish. Then the relation between the quantities p ik and 
x i is one-to-one, and x i may be used as the coordinates of a line. 

By the substitution (1) the fundamental relation &>(r)= goes 
into a quadratic relation of the form 

^(x)=^a ik ^x k = 0. <>,,= % ) (2) 

In fact, by a proper choice of the coefficients in (1), the function 
f (V) may be any quadratic form of nonvanishing discriminant and, 
in particular, may be a sum of the six squares x\. The proof of 
this may be given as a generalization of the similar problem in 
space or may be found in treatises on algebra, 

By the substitution (1) the polar a>(r, r f ) goes into the polar 



To prove this let r ik and r f ik represent two sets of values of the coor 
dinates r ik and let x i and x\ represent the corresponding values of the 
coordinates x then r ik +\r f ik corresponds to # f -h\2:( for all values of \. 

Therefore a> (r + X/) = (a + Xa/), 

or o> (r) + 2 Xo> (r, r ) + X 2 o> (r ) = (af) + 2 Xf (z, a/) + X 2 



306 FOUR-DIMENSIONAL GEOMETRY 

By equating like powers of X we have 

(r, r )=fO>*0- 

Hence the ratios of any system of six quantities #., bound by a 
homogeneous quadratic relation % (x) = of nonvanishing discrimi 
nant, may be taken as the coordinates of a line in space in such a 
manner that the equation (x, a/) = is the necessary and sufficient 
condition for the intersection of the two lines x { and x . 

Of particular importance are coordinates due to Klein, to 
which we shall refer as Klein coordinates. These are obtained by 
the substitution 



The fundamental relation is then 

x* + x? + x* + xl + xl + xl = 0, 
and the condition for the intersection of two lines is 



131. Pencils and bundles of lines. /. If a i and b i are two inter 
secting lines, then px { a { -\- \b { is a line of the pencil determined by 
a f and b { , and any line of the pencil may be so expressed. 

The hypotheses are 

00=0, 



Then: 

1. x i are the coordinates of a straight line, since 



2. The line x i lies in the plane of a { and b i and passes through 
their point of intersection. To prove this let d { be any line cutting 
both a f and b r That is, d { is either a line through the intersection 
of a i and 6 t . or a line in the plane of a. and b r Then f (a, d) = 0, 
and f (b, d) = 0. Therefore 



LINE COORDINATES 307 

Hence x i intersects any and all of the lines d. and therefore lies 
in the plane of a i and b t and passes through their intersection. 

3. The value of X may be so taken as to give any line of the 
pencil determined by a. and b^ To prove this let P be any point 
of the pencil except its vertex, and let h { be a line through P but 
not in the plane of a i and b r We can determine X so that 
{(x, A)=f(o, A) + X(6, A)=0. 

Hence x. intersects h t ; and since 7i i has only the point P in the 
plane of a t and b { , and x i lies in that plane, x i passes through P and 
is any line of the pencil. The theorem is completely proved. 

//. If a tJ 5., and c. are three lines through the same point but not 
belonging to the same pencil, then px { = a t + \b.-{- pc. is a line through 
the same point, and any line through that point may be so represented. 

By hypothesis, f (a) = 0, f (ft) = 0, f (c) = 0, f (a, 6) = 0, ? (6, c) = 0, 
f(c, a)=0. Then: 

1. x i are the coordinates of some line, since f (V) = 0- 

2. Any line which cuts all three lines a { , 6 { , and c { cuts #,.. For, 
if f(o, d)=0, f(5, d)=0, and f(c, rf) = 0, then f(>, tf)=f(a<f) 
+ Xf (5, ?)+/*(<?, <?) = 0. Therefore ^ passes through the inter 
section of a^ b t , c { . 

3. Values of X and p may be so determined that x i may cut 
any two lines g i and A t . which do not cut the lines a { , 6 { , and c t .. We 
have, in fact, to determine X and ft from the two equations 

#) + Xf (6, #)+ /x|(c, #)= 0, 
A) + Xf (6, A) 4- /if (c, A) = 0. 
The theorem is therefore proved. 

///. If t ., ., and Cf are any three lines in the same plane but not 
belonging to the same pencil, then px { a { -\- X5 f -f- fJLc i is a line in the 
same plane, and any line in the plane may be so represented. 

The proof is the same as for theorem II. 

A configuration consisting of all lines through the same point 
is called a bundle of lines. A configuration consisting of all lines 
in a plane is a plane of lines. By the use of line coordinates we 
do not distinguish between a bundle and a plane of lines. In fact 
each configuration consists of a doubly infinite set of lines each of 
which intersects all of the others. 



308 FOUR-DIMENSIONAL GEOMETRY 

EXERCISES 

1. Prove that the cross ratio of the four points in which a straight 
line meets the four planes of any tetrahedron is equal to the cross 
ratio of the four planes through the line and the vertices of the 
tetrahedron. 

2. Prove that there are two and only two lines which intersect four 
given lines in general position. 

3. Prove that if the coordinates of any five lines satisfy the six 
equations te, + W + w, + ,., + <*, = <* 

the five lines intersect each of two fixed lines. 

4. Show that if the coordinates of any four lines satisfy the six 
equations n 

\X t + fjUJf + VZi + pS { = 0, 

any line which intersects three of them intersects the fourth, and hence 
the lines are four generators of a quadric surface. 

5. Show that if the coordinates of three lines are connected by the 

six equations . A 

\Xi + fii/i -f vZf = 0, 

any line which intersects two of them intersects the third. Thence 
deduce that the lines are three lines of a pencil. 

\ 132. Complexes, congruences, series. A line complex is a three- 
dimensional extent of lines. It may be, but is not necessarily, 
defined by a single equation which is satisfied by the coordinates 
of the lines of the complex. The order of a complex is the num 
ber of its lines which lie in an arbitrary plane and pass through 
an arbitrary point of the plane ; that is, it is the number of the 
lines of the complex which belong to an arbitrary pencil. 

A line congruence is a two-dimensional extent of lines. It may 
be defined by two simultaneous equations in line coordinates and 
is then composed of lines common to two complexes. The order 
of a congruence is the number of its lines which pass through an 
arbitrary point ; its class is the number of its lines which lie in an 
arbitrary plane. 

A line series is a one-dimensional extent of lines. It may be 
defined by three simultaneous equations in line coordinates. It 
then consists of lines common to three complexes. The order of a 
series is the number of its lines which intersect an arbitrary line. 



LINE COORDINATES 309 



An equation /0*v x & x tf x s x tf x *) ~ ^ 

where / is a homogeneous polynomial of the nth degree in x 9 
defines a line complex of the nth order. Let a i and b i be any two 
fixed intersecting lines. Then a f + \&. is, by theorem I, 131, a line 
of the pencil denned by a i and >., and this line will belong to the 
complex (1) when X satisfies the equation 

/(! + X^, a 2 + X6 2 , a g + X6 3 , 4 + X 4 , 5 + X 5 , 6 + X5 6 )= 0, 

which is of the wth degree in X. 

From the above it follows that through any fixed point of space 
goes a configuration of lines such that n of these lines lie in each 
plane through the fixed point. Since the relation between the 
coordinates of the fixed point and those of any point on a line 
of the complex is an analytic one, derived from (1), it follows 
that any point of space is the vertex of a cone of nth order formed 
~by lines of the complex. 

Also if we consider a fixed plane, through every point of it go 
n lines of the complex. Since, as before, we have to do with an 
analytic equation, we infer that in any plane the lines of a complex 
envelop a curve of the nth class. 

A simple example of a line complex is that which is composed 
of all lines which intersect a fixed line. For if a i are the coordi 
nates of a fixed line A, the condition that a line x i should intersect 
A is, by 130, f (.*)=. (2) 

which is a linear equation. Hence this complex is of the first 
order. In fact through an arbitrary point in an arbitrary plane 
goes obviously only one line intersecting A. Through a fixed point 
M goes a pencil of lines ; namely, the lines through M in the plane 
determined by M and A. This is a cone of the first order. In any 
plane m goes a pencil of lines ; namely, the lines through the point 
in which m intersects A. These form a line extent of the first class. 

Another example of a line complex is one of second order 
defined by the equation 

pi* + A + pi +A +pl* + A = o. 

which, expressed in point coordinates, is 



310 FOUK-DIMENSIONAL GEOMETRY 

This is not the equation of a surface, since it contains two sets 
of point coordinates. If, however, the coordinates y { are fixed, 
(4) becomes the point equation of the cone of second order formed 
by lines of the complex through y.. 

If, dualistically, we express equation (3) in plane coordinates w t 
and v i and hold v { fixed, we obtain a plane extent of second class 
in U+ which is intersected by the plane v i = const, in a line extent 
enveloping a curve of second class. 

Through an arbitrary point in an arbitrary plane go two lines 
of the complex (3). 

An example of a line congruence is that of lines intersecting 
two fixed lines. It is represented by two simultaneous equations 
similar to (2). It is of the first order, since through any point 
but one line can be passed intersecting the two fixed lines. It is 
of -soo^nH class, since in a fixed plane only one line can be drawn 
intersecting the two fixed lines. 

Another example of a line congruence consists of all lines through 
a point. This is of first order and zero class. Still another example 
consists of all lines in a plane. This is of zero order and first class. 

An example of a line series is that of lines which intersect three 
fixed lines and is represented by three linear equations of the 
form (2). Such lines are one family of generators on a surface of 
second order ( 96). The series is of $eee4- order, since any line 
in space meets two lines of the series/"" 

133. The linear line complex. The equation 



a i x i 



where x t are general line coordinates, defines a linear line complex. 
An example of such a complex is, as we have seen, that which is 
composed of lines cutting a fixed line. Such a complex we call a 
special linear line complex or, more concisely, simply a special complex. 
The necessary and sufficient condition that (1) should represent a 
special complex is that the equation (1) should be equivalent to 



that is, that p* < = , (2) 



where y i are the coordinates of a -point and therefore satisfy the 
equation fO)=0- (3) 



LINE COORDINATES 



311 



Equations (2) can be solved for y, since the discriminant of (3) 
does not vanish ( 130). The results of the solution substituted 
in (3) give a relation of the form 

,00=0, (4) 

where rj (#) is a homogeneous quadratic polynomial in a;.. 
We sum up as follows : 

/. A special linear complex is composed of straight lines which 
intersect a fixed line called the axis of the complex. A linear equa 
tion (1) defines a special complex when and only when the coefficients 
a i satisfy the quadratic equation (4). 

More in detail, let 

f GO =2X&& (= a ) (5) 

Then equations (2) are 

P a n (6) 

0. (7) 



from which, together with (5), we have 



From (6) and (7) we obtain 



17 (a) = 



where A ik is the cofactor of a ik in the expansion of D = \ a ik |. 

Then = A n < 
" D 



the last result coming from the solution of equations (6) for 
If we have Klein coordinates 

^() = a* + al + < 4- <*l + <+ <, 



312 FOUK-DlMMSIONAL GEOMETRY 

We may sum this up in the following theorem : 

//. The coordinates of the axis of the complex (1) when it is special 

are If Klein coordinates are used, the coordinates of the axis of, a 
da t 

special complex are the coefficients in the equation of the complex. 

Returning to the general linear complex (1) (special or non- 
special), consider any point P. If a fl b# and c i are any three lines 
through P not in the same plane, then (theorem II, 131) any 
line through P has coordinates a f +X& t .+ /-w? t , and this line belongs 
to the complex when 



Equation (8) is satisfied for all values of X and /JL if the three 
lines a f , b { , and c t belong to the complex. Otherwise, assuming 
that c t . does not belong to the complex, we may solve (8) for p 
and write the coordinates of the poiat x { in the form 



where a\ and b[ are two definitely defined lines through P, and X 
is arbitrary. This proves the following theorem: 

///. Through any arbitrary point in space goes a pencil of lines of 
the complex unless in an exceptional manner all lines through the point 
belong to the complex. 

The analysis would be the same if the three- lines a., >., and c i 
were taken as three lines in a plane, but not through the same 
point (theorem III, 131). Hence 

IV. In any arbitrary plane in space lies a pencil of lines of the 
complex unless in an exceptional manner all lines of the plane belong 
to the complex. 

To complete the information given by these two theorems we 
shall prove the two following: 

V. If all lines through any one point P belong to the complex, the 
complex is special and the point P lies on the axis of the complex. 

Let all lines through P (Fig. 56) be lines of the complex. Take 
h, a line not belonging to the complex, and let Q and It be two 



LINE COORDINATES 



313 



R 




\ ! 



FIG. 56 



points of h. Through Q goes, by theorem III, a pencil of lines of 

the complex of which PQ is evidently one and h is not. Similarly, 

through R goes a pencil of lines of the complex of which RP is 

one and h is not. These two pencils lie in different planes, for if 

they lay in the same plane the line 

h would lie in both pencils and 

be a line of the complex, contrary 

to hypothesis. The planes of the 

pencils intersect in a line which 

contains P. Call it c, and let S be 

any point on c. 

The line SP belongs to the com 
plex, since, by hypothesis, all lines 
through P are lines of the complex. 
The line SQ belongs to the com 
plex, since it lies in the plane of the pencil with the vertex Q and 
passes through Q. Similarly, the line SR belongs to the complex. 

Therefore we have, through the point S, three lines of the 
complex which are not coplanar, since c and h are not in the 
same plane. Hence, by theorem III, all lines through S belong to 
the complex. But S is any point of , and since all lines which 
intersect c form a complex, the 
theorem is proved. 

VI. If all lines of a plane be 
long to the complex, the complex 
is special and the plane passes 
through the axis of the complex. 

Let all lines of a plane m 
(Fig. 57) belong to the com 
plex. Take A, any line not of 
the complex, and let q and r be 
two planes through A, intersect 
ing m in the lines mq and mr. In the plane q lies, by theorem IV, 
a pencil of lines of the complex of which mq is one and h is not. 
Similarly, in the plane r lies a pencil of lines of the complex 
of which mr is one and h is not. These pencils have different 
vertices, for otherwise they would contain h. Let c be the line 




FIG. 57 



314 FOUR-DIMENSIONAL GEOMETRY 

connecting the vertices (e, of course, lies in m). Take s, any plane 
through c intersecting q in the line qs and r in the line rs. 

Then c is a line of the complex, since by hypothesis any line in 
m belongs to the complex. Also qs and rs belong to the complex, 
since each is a line of a pencil which has been shown to be com 
posed of lines of the complex. The three lines do not pass through 
the same point because qm and rm have been shown to intersect c 
in different points. 

Therefore, by theorem IV, all lines in * belong to the complex, 
and since s was any .plane through c, all lines which intersect c 
belong to the complex, and the theorem is proved. 

134. Conjugate lines. Two lines are said to be conjugate, or re 
ciprocal polars, with respect to a line complex when every line of 
the complex which intersects one of the two lines intersects the 
other also. Let the equation of the complex in Klein coordinates be 

a i x i + V* + V* + "4*4 + Vs + Ve = C 1 ) 

and let y { and z. be the coordinates of any two lines. The condi 
tions that a line x. intersect y. and z i are respectively 



We seek the condition that any line x i which satisfies (1) and (2) 
will satisfy (3). This condition is that a quantity X shall be found 

such that s- -i o o <~ n^ 

P*i= &+ *> a i- (* =1 2 3 ^ ^ 6 ) (4) 

But y { and z i both satisfy the fundamental relation 



Therefore, from (4), \ = - -=^~ , (5) 

2Ya.# t . 
and (4) becomes pz i y i ~* * *g t ., (6) 

5X 

which define the coordinates z i of the conjugate line of any line y t . 
From (5) follows at once the theorem : 

/. Any line has a unique conjugate with respect to any nompecial 
complex. 



LINE COORDINATES 315 

If the line y i belongs to the complex, then Va.?/.= and pz. = y.. 
Hence 

//. Any line of a nonspecial complex is its own conjugate. 

If the complex is special, ^P *=(). Therefore, unless also 
V .?/.= 0, \ = oo and pz { = a r Hence 

///. The axis of a special complex is the conjugate of any line not 
belonging to the complex. 

If the complex is special and the line y. belongs to it, X is 
indeterminate. Hence 

IV. A line of a special complex has no determinate conjugate. 

The above theorems may also be proved easily by purely geo 
metric methods. 

If two lines have coordinates y { and z { which satisfy equations (6), 
then any values of x i which satisfy (2) and (3) will also satisfy (1). 
Hence 

V. If two lines are conjugate with respect to a complex, any line 
ivhich intersects both of them belongs to the complex. 

From this theorem or from the relations (6) follows at once : 

VI. Two lines .conjugate with respect to a nonspecial complex do not 
intersect. 

We have seen (theorem IV, 133) that in any plane m there is 
a unique point P which is the vertex of the pencil of complex 
lines in m. Similarly, through any point P goes a plane m which 
contains the pencil of complex lines through P. When a point and 
plane are so related, the point is called the pole of the plane, 
and the plane is called the polar of the point. 

If g and h are two conjugate lines with respect to a complex, 
and P is any point on g, the pencil of lines from P to points 
on h is made up of complex lines by theorem V. Hence follow 
the theorems: 

VII. The polar plane of a point P on a line g is the plane deter 
mined by P and the conjugate line h. As P moves along g the polar 
plane turns about h. 

VIII. The pole of any plane m through a line g is the intersection of 
m with the conjugate line h. As m turns about g its pole traverses h. 



316 



FOUR-DIMENSIONAL GEOMETRY 



135. Complexes in point coordinates. It is interesting and instruc 
tive to consider the linear complex with the use of point coordinates. 
A linear equation in general line coordinates 



is equivalent to a linear equation 



in p ik coordinates, and this, again, can be expressed as a bilinear 
equation in point coordinates : 



If in equation (3) we place y { equal to constants, the equation 
becomes that of a plane m of which y. is the pole. 
The plane coordinates of this plane are 



(4) 



and to each point y i corresponds a unique plane unless 







<ia i. 14 



12 23 ,4 

- -" M o 

- M - 



= 0; 



that is, unless (a 12 34 4- 13 42 + a 14 M ) a = - 



But tf-f ^ 



fo rm which 77(0;) takes for the p i 



coordinates. Hence we have a verification of the fact that in a 
nonspecial complex any plane has a unique pole. 

Let us take two conjugate lines as the edges AB (x^= 0, x 2 = 0), 
and CD (# 8 = 0, x^= 0) of the tetrahedron of reference for the point 
coordinates. This can always be done by a collineation which 
obviously amounts to a linear substitution of the line coordinates. 

If : : y 3 : y^ is a point P on AB, its polar plane is, by (3), 



LINE COORDINATES 317 

This plane must pass through CD for all values of y z and y f 
Hence a 13 = a u = a 42 = # 23 = 0, and the line complex reduces to 



where neither of the coefficients can be zero if the complex is 
nonspecial. 

It is possible to make the ratio a u : a 34 equal to 1 by a colline- 
ation of space. To see this, note that if we place 



then p[ 2 = a i 2 Pi2-> an d / > 34 = ~~ a 34/ ) 34 an d ^ ne equation of the com- 

plexbecomes 



Consider now a special complex, and let its axis be taken as 
the line AB (2^=0, x 2 = 0), the line coordinates of which are 
Piz ~ Pis Pu j42 = ^23 ^- The condition that a line should inter 
sect this line is, by (1), 129, 



We may sum up in the following theorem : 

By a projective transformation of space the equation of any special 
complex may be brought into the form 

p 12 = 

and that of any nonspecial complex into the form 



136. Complexes in Cartesian coordinates. We shall now consider 
the properties and equations of line complexes with the use of 
Cartesian coordinates x-.y:z:t, by which the plane at infinity is 
unique and metrical properties come into evidence. 

For special complexes we have two cases, according as the axis 
is or is not at infinity. In the former case the lines which inter 
sect it are parallel to a fixed plane. Hence 

In Cartesian geometry the special line complex consists either of all 
lines which intersect a fixed line or of all lines which are parallel to a 
fixed plane. 



318 FOUR-DIMENSIONAL GEOMETRY 

Consider a nonspecial complex. In the plane at infinity is a 
unique point I, the pole of the plane. The lines of space which 
pass through I form a set of parallel lines not belonging to the 
complex. These are called the diameters of the complex. Each 
diameter is conjugate to a line at infinity, since the conjugate to a 
diameter must meet all the pencil of lines of the complex whose 
vertex is I. Conversely, any line at infinity not through I has a 
diameter as its conjugate. In other words, the polar planes of points 
on a diameter are parallel planes, and the poles of any pencil of paral 
lel planes lie on a diameter. 

Consider now the pencil of parallel planes formed by planes 
which are perpendicular to the diameters. Their poles lie in a 
diameter which is unique. Therefore there is in each nonspecial 
complex a unique diameter, called the axis, which has the property of 
being perpendicular to the polar planes of all points in it. 

Referring to (4), 135, if we replace xjxjxjx^ by x\y\z\t, 
the pole of the plane at infinity is given by the equations 



which have the solution 

x:y:z:t = a 2S :-a ls :a l2 :Q. (1) 

Any line through the point (1) is therefore a diameter, and if 
0*V y\"> z i) * s an y nn ite point of space, the equation of the diameter 
through it is 



x-x^ = y-y^ = z-z^ 

a <x ~ a u ~ a n 

The polar plane of (x^, y^, z^) is, by (4), 135, 



The line (1) is perpendicular to the plane (2) when 






Consequently, if (x^ y^ z^) in (3) are replaced by variable 
coordinates (x, y, z), equation (3) becomes the Cartesian equation 
of the axis of the complex. 



LINE COORDINATES 319 

Let us take this axis as the axis of z. Then, from (1), 2g = 0, 
a w~ an d> from (3), since the origin of coordinates is on the axis, 
a u = a 42 = The equation of the complex is then 



which agrees with (5), 135. 

In Cartesian coordinates equation (4) is 

xy -x y + k(z-z ) = Q, (5) 

which associates to any point (V, y , z ) its polar plane. 

From (5) it is obvious that the polar plane of P (x , y , z ) 
contains the line xy x y = Q, z-=z , which is the line through 
P perpendicular to the axis. The normal to the plane makes with 

k Va/ 2 -|- y 12 r 

the axis the angle cos" 1 - = tan" 1 - y = tan" 1 - 

Vz 2 +y 2 + & 2 k k 

where r is the distance from P to the axis. This leads to the 
following result: 

The polar plane of any point P contains the line through P 
perpendicular to the axis. If P is on the axis, its polar plane is per 
pendicular to the axis. As P recedes from the axis along a line 
perpendicular to it, the normal plane turns about this perpendicular, 
the direction and amount of rotation depending upon the sign and the 
value of k. If P moves along a line parallel to the axis, its polar 
plane moves parallel to itself. 

Any line of a complex may be denned by a point (#, y, z) and 
its neighboring point (x + dx, y + dy, z + dz). If in (5) we place 
x = x + dx, y = y + dy, z = z -f- dz, we have 

xdy ydx kdz = 0, (6) 

which may be called the differential equation of the complex. 

Equation (6) is of the type called nonintegrable, in the sense 
that no solution of the form f(x, y, z, c) = can be found for it. 
It is satisfied, however, in the first place, by straight lines whose 

equations are 

z = c, y = mx. (7) 

In the second place, on any cylinder with the equation 

^+/=a 2 (8) 

may be found curves whose direction at any point satisfies (6). 



320 FOUR-DIMENSIONAL GEOMETRY 

For the direction of any curve on (8) satisfies the equation 



and this equation combined with (6) gives the solution 

Q 

(9) 



K X 

2 7T& 2 

which are the equations of helixes with the pitch . 

K 

It appears from the preceding that any tangent line to a helix 
of the form (9) is a straight line of the complex. We shall now 
prove, conversely, that any line of the complex, excepting only 
the lines (7), is tangent to such a helix. 

Since z is assumed not to be constant, we may take the equation 
of any line not in the form (7) as 

x = mz + 5, y = nz+p, (10) 

with the condition bn pm = &, which is necessary and sufficient 
in order that equations (10) should satisfy (6). 

The distance of a point (x^ y^ z^) on (10) from OZ is 



2 (mb + np) *! + 
It is easily computed that this distance is a minimum when 

mb 4- np nk mk 

~ ~ 



k 
The minimum distance is == , which we shall take as a in 

Vw 2 +7i 2 

the equations of the helix (9). The direction of the helix at the 
point (^, y^ z^) is ^ 

dx : dy : dz = y^. x^: = m : n : 1. 

This is the direction of the line (10), and our proposition is proved. 
We have, therefore, the following theorem : 

A linear nonspecial complex may be considered as made up of the 
tangents to the helixes drawn upon cylinders whose axes coincide with 

2 7T# 2 

the axis of the complex, the pitch of each helix being - , where a is 

K 

the radius of the cylinder and k the parameter of the complex. 



LINE COORDINATES 321 

137. The bilinear equation in point coordinates. The equation 

2>**<y*=o (i) 

is the most general equation which is linear in each of the two 
sets of point coordinates (x^i x 2 : x s : # 4 ) and (y^-y^y^y^)* 

By means of (1) a definite plane is associated to each point 
y^ its equation being obtained by holding y i constant in (1). 
Similarly, to each point x i is associated a definite plane. 

In this book we have met two important examples of equation (1). 

I. a ki = a ik . Equation (1) then associates to each point y { its 
polar plane with respect to the quadric surface 



The pole does not in general lie in its polar plane. Exceptions 
occur only when the pole is on the quadric. 

II. a ki = a ik \ whence a n = 0. Equation (1) associates to each 
point y { its polar plane with respect to the line complex 



The point y { always lies in its polar plane. This association 
of point and plane forms a null system, mentioned in 102, and here 
connected with the line complex. 

EXERCISES 

1. Prove that a complex is determined by any five lines, provided 
that they are intersected by no line. 

2. Prove that a complex is determined by a pair of conjugate lines 
and any line not intersecting these two. 

3. Prove that a complex is determined by two pairs of conjugate lines. 

4. Prove that if a line describes a plane pencil its conjugate also 
describes a plane pencil, and if a line describes a quadric surface its 
conjugate does also. 

5. Prove that a complex (or null system) is in general determined by 
any three points and their polar planes. 

6. Prove that any two pairs of polar lines lie on the same quadric 
surface. 

7. Prove that the conjugate to the axis of a nonspecial complex is 
the polar with respect to the imaginary circle at infinity of the pole of 
the plane at infinity with respect to the complex. 



322 FOUR-DIMENSIONAL GEOMETRY 

138. The linear line congruence. Two simultaneous linear equa 
tions in line coordinates, 

=0, (1) 

define a congruence. Evidently equations (1) are satisfied by all 
lines common to two linear complexes. But all lines which belong 
to the two complexes defined by equations (1) belong also to 
all complexes of the pencil 

(X.+ Xft) *,= <), (2) 

and the congruence can be defined by any two complexes obtained 
by giving X two values in (2). 

A complex defined by (2) is special when 



that is, when rj (a) + 2 XT; (a, ) + \ 2 rj () = 0. (3) 

In general equation (3) has two distinct roots. Hence we have 
the theorem: 

In general the linear congruence consists of straight lines which 
intersect two fixed straight lines. 

The two fixed lines are called the directrices of the congruence. 
The directrices are evident conjugate lines with respect to any 
nonspecial complex defined by equation (2). 

If the roots of equation (3) are equal, the congruence has only 
one directrix and is called a special congruence. This congruence 
consists of lines which intersect the directrix and also belong to 
a nonspecial complex. It is clear that the directrix must be a 
line of this nonspecial complex, for otherwise it would have a 
conjugate line and the congruence would be nonspecial. Hence 
a special congruence consists of lines which intersect a fixed line and 
such that through any point of the fixed line goes a pencil of con 
gruence lines, the fixed line being in all cases a line of the pencil. 

As the vertex of the pencil moves along the directrix, the plane 
of the pencil turns about the directrix. 

We have seen that a nonspecial congruence may be defined by 
its directrices. If the directrices intersect, the congruence separates 



LINE COORDINATES 323 

into two sets of lines, one being all lines in the plane of the direc 
trices (a congruence of first order and zero class), and the other 
being all lines through the point of intersection of the directrices 
(a congruence of zero order and first class). 

When the directrices do not intersect, the congruence is one of 
first order and first class. 

139. The cylindroid. We have seen that every linear complex has 
an axis. In a pencil of linear complexes given by equation (2), 
138, there are, therefore, <x> l axes which form a surface called 
a cylindroid. We may find the equation of the cylindroid in the 
following manner: 

Let us take as the axis OZ the line which is perpendicular to 
the directrices of the two special complexes of the pencil, as 
the origin the point halfway between the two directrices, as the 
plane XOY the plane parallel to the two directrices, and as OX 
and OY the lines in this plane which bisect the angles between 
the two directrices. That is, we have so chosen the axes of refer 
ences that the equations of the two directrices of the special 
complexes of the pencil are 

y-mx=0, z = c 9 (1) 

and y -f mx = 0, z = c, (2) 

respectively. 

The Pliicker coordinates of the line (1), which may be deter 
mined by the points (0, 0, c) and (1, m, c), are 

*,(!)__ m (1 >-t nM 1 -n^ 1 ) nmt* vP> nm ^(D 
Pl2 v Pis ~ L i Pl4 *1 Pv& mc< > P*2 m Ps* U > 

and the special complex with this axis is therefore, by (1), 129, 
mp 13 - mcp u -p 23 - cpu = 0. 



Similarly, the coordinates of (2) are 

-/a) _ (2) c (2) 1 rP } mc > (2) ra (2) 
Pi-i v, Pis c i Pu -M Pzs mc i Ptz <"> Pwv-) 

and the special complex with this axis is 

- mp u - mcpu - p 2B + cpu = 0. 
The pencil of complexes is therefore 

(1 - X) mp u - (1 + X) mcp u - (1 + \)p M $ (1 - X) cp 4 , = 0. 



324 FOUR-DIMENSIONAL GEOMETRY 

By (3), 136, the equations of the axis of any complex of the 
pencil are 



X) 




which reduce to y = ^ - mx, 

-L -p A, 



[(1 - X)W + (1 + X) 2 ] g = (1 - X 2 ) (1 + w 2 ) g. 
If we eliminate X from these equations, we have 

c ^ = 0, (3) 



which is the required equation of the cylindroid. 

The equations show that the surface is a cubic surface with OZ 
as a double line. All lines on the surface are perpendicular to OZ, 
and in any plane perpendicular to OZ there are two lines on the 
surface which are distinct, coincident, or imaginary according as 
the distance of the plane from is less than, equal to, or greater 

(l + w 2 )c 
than ^ - - ^-. 
2m 

We may put the equation of the cylindroid in another form. We 
shall denote by 2 a the angle between the directrices of the special 
complexes of the pencil, by the angle which any straight line 
on the cylindroid makes with OX, and by r the distance of that 

line from 0. Then m = tan a, and - - m = tan 6. 
Equation (3) then becomes 

_ sin 20 



sin 2 a 



140. The linear line series. Consider three independent linear 
equations _ Q> ^ ^ .^ 0. 



These equations are satisfied by the coordinates of lines which 
are common to the three complexes defined by the individual 
equations in (1) and define a line series. Any line of the series 
also belongs to each complex of the set given by the equation 



LINE COORDINATES 325 

and any three linearly independent equations formed from (2) by 
giving to X, /A, and v definite values determine the same line series 
that is determined by (2). 

A complex of the type (2) is special when 
77 (\a -f- fji/3 -+- 1/7) = X 2 77 O) H- //, 2 77 (0) + z> 2 77 (7) + 2 X/A77 (#, 0) 

+ 2 w (0, 7) 4- 2 ^77 (7, a) = 0. (3) 

There are a singly infinite number of solutions of equation (3) 
in the ratios X : //, : v. Hence the lines which are defined by equa 
tions (1) intersect an infinite number of straight lines, the axes 
of the special complexes defined by (2) and (3). These lines are 
called the directrices. 

The arrangement of the directrices depends upon the nature of 
equation (3). In studying that equation we may temporarily in 
terpret X : fji : v as homogeneous point coordinates of a point in a 
plane and classify equation (3) as in 35. 

Let us place , , , . 

77 O) 770, 0) 770,7) 

770,7) 77(0,7) 17(7) 

CASE I. D =#= 0. This is the general case. Equation (3), inter 
preted as an equation in point coordinates X : p : v, is that of a conic 
without singular points. To any point on this conic corresponds a 
special complex of the type (2) whose axis is a directrix of the 
series (1). To simplify our equations we shall assume that the 
coordinates x { are Klein coordinates. Then (by theorem II, 133) 
if (X 1 : ^ : i/j) and (X 2 : /* 2 : i> 2 ) are two solutions of equation (2), 
the axes of the corresponding special complexes, or, in other words, 
the corresponding directrices of the series (1), are X^-f- A t 1 t + v t y. 
and X;2#j-|- /x 2 0j-f- v<fYi. 

The condition that these two directrices intersect is 



which is exactly the same as the condition that each of the two points 
(X x : p l : i/ t ) and (X 2 : ft 2 : v^) should lie on the polar of the other with 
respect to the conic (3). This is impossible, since each of the points 
lies on the conic. It follows from this that no two directrices intersect. 
From this it will also follow that no two lines of the given series 
intersect, for if they did each directrix must either lie in their 



326 



FOUK-DIMENSIONAL GEOMETRY 



plane or pass through their common point, and some of the 
directrices would intersect. 

The lines of the series (1), on the one hand, and their direc 
trices, on the other, form, therefore, two families of lines such that 
no two lines of the same family intersect, but each line of one 
family intersects all lines of the other. This suggests the two fam 
ilies of generators on a quadric surface. That the configuration 
is really that of a quadric surface follows from the theorem that 
the locus of lines which intersect three nonintersecting straight 
lines is a quadric surface (see Ex. 6, p. 327). 

We sum up in the following words : 

In the general case (Z> = 0) the lines which are common to three 
linear complexes form one family of generators of a quadric surface, 
their directrices forming the second family. 

A family of generators of a quadric surface is called a regulus. 

CASE II. D = 0, but not all the first minors are zero. The curve 
of second order defined by (3) reduces to two intersecting straight 
lines and, by a linear substitution, can be reduced to the form 

X/4-0. 

To do that we must define the series by three complexes such that 
0, *?(>)= > iK&,<0=0, i/O, = 0, 17 (,)*=<). 




These are three special com 
plexes such that the axes of 
the first two do not intersect, 
but the axis of the third inter 
sects each of the axes of the 
first two. The axes lie, there 
fore, as in Fig. 58. The series 
consists, therefore, of two pen 
cils of lines : one lying in the 
plane of a and c, with its vertex at F 1 , the point of intersection 
of b and c ; the other lying in the plane of b and c, with its vertex 
at F, the intersection of a and c. 

CASE III. D = 0, all the first minors are zero, but not all the 
second minors are zero. The conic defined by (3) consists of two 
coincident lines. Its equation may be made v 2 = 0. 



FIG. 58 



LINE COORDINATES 327 

We have then taken to define the series three complexes of 
which two are special- with intersecting axes, and the third is non- 
special and contains the axis of the other two. 

If a and b are the two axes of the special complexes, F their 
point of intersection, and m their common plane, then, since the 
nonspecial complex contains a and 6, F is the pole of m with 
respect to that complex. Hence the lines common to the two 
complexes form a pencil of lines which must be taken double to 
preserve the order of the complex. 

CASE IV. The case in which all the second minors of D vanish is 
inadmissible, for in that case the three complexes in (1) are special 
and their axes intersect. Then, from 131, y.= a t + v/3 i9 and the 
three equations (1) are not independent. 

EXERCISES 

Two complexes f ajc i = Q and ^.bjK i = are in involution when 



1. Prove that if p is a line common to two complexes in involution 
the correspondence of planes through p, which can be set up by taking 
as corresponding planes the two polar planes of each point of p with 
respect to the two complexes, is an involution. 

2. Prove that two special complexes are in involution when their 
axes intersect. 

3. Prove that a special complex is in involution with a nonspecial 
complex when the axis of the former is a line of the latter. 

4. Prove that if two nonspecial complexes are in involution there 
exist two lines, g and /*, which are conjugate with respect to the two 
and such that the polar planes of any point P are harmonic conjugates 
with respect to the two planes through P and g and through P and h 
respectively, and also such that the poles of any plane in with respect 
to the two complexes are harmonic conjugates to the points in which m 
meets g and h. 

5. Prove that the six complexes a\- = 0, where x { are Klein coordi 
nates, are two by two in involution. Hence prove by a transformation 
of coordinates that there exists an infinite number of such sets of six 
complexes mutually in involution. 

6. Prove that the locus of lines which intersect three nonintersecting 
lines is a quadric surface, by using Pliicker coordinates and eliminating 
one set of point coordinates. 



328 FOUR-DIMENSIONAL GEOMETRY 

141. The quadratic line complex. A quadratic line complex is 
defined by an equation of the form 



We shall consider only the general case in which the above 
equation can be reduced to the form 



at the same time that the coordinates x i are Klein coordinates 
satisfying the fundamental relation 

2>HO. (2) 

Let us consider any fixed line y i of the complex and any linear 
com P lex 



containing y^ In general the complex (3) will have two lines 
through any point P in common with (1), for P is at the same 
time the vertex of a pencil of lines of (3) and of a cone of lines 
of (1). 

Analytically, we take P, a point on y { , and z { , any line of (3), 
but not of (1), through P. Then any line of the pencil determined 

by y. and z. is , ,. 

W^Jfi+Xjyi 

and this line always belongs to (3), but belongs to (1) when and 
only when 



This gives in general two values of X, of which one, X = 0, deter 
mines the line y { and the other determines a different line. But 
the two values of X both become zero, and the line y t is the only 
line through P common to (1) and (3) when 



that is, when z. has been chosen as any line of the linear complex 

5) W<= 0. (4) 

In this case the polar plane of P with respect to (4) is tangent to 
the complex cone of (1) at P, where P is any point whatever of y { . 
The complex (4) is accordingly called the tangent linear complex 
at y ( . It is often said that the tangent linear complex contains all 



LINE COORDINATES 329 

lines of the complex (1) which are consecutive to y^ since any line 
with coordinates y { + dy i satisfies (4). The discussion we have given 
makes this notion more precise. 

More generally we have at y { a pencil of tangent linear com 
plexes. For by virtue of (2) the complex (1) may be written 



0, (5) 

where /JL is any constant, and the tangent linear complex to (5) is 

0. (6) 



All these complexes have the same polar plane at any point P of y { . 

If y. is not a line of the complex, equation (6) defines a pencil 
of polar linear complexes. 

The line # t . is called a singular line when the tangent linear 
complex (4) is special. The condition for this is 



?= 0, (7) 

which says that c i y i are the coordinates of a line, the axis of the 
tangent complex. At the same time all the complexes (6) are special 
and have the same axis. 

This axis intersects y^ since ^<?#J= (because y. is a line of the 
complex), and the intersection of the two lines is called a singular 
point, and their plane a singular plane. Any complex line y. for 
which condition (7) holds is called a singular line. 

Let P be a singular point on a singular line y it let z. be any line 
through P, and consider the pencil of lines 

/>*,.= &+**< (8) 

The condition that x i belong to (1) is 

0, (9) 

since c i yf = 0, because y i is on (1), and ]} <?,#&= 0, because z. 
intersects c i y i at P. Then if z. is a line of (1), all lines of the pencil 
(8) belong to (1). On the other hand, if z i is any line not belonging 
to the complex (1), the line y. is the only line in the plane (j/<2 t ) 
which belongs to the complex. This makes it evident that at a 
singular point the complex cone splits up into two plane pencils 
intersecting in the singular line. 



330 FOUR-DIMENSIONAL GEOMETRY 

In a similar manner we may take p as a singular plane through 
a singular line y^ z { , any line in p intersecting y { , and again con 
sider the pencil (8). We obtain again (9), but the interpretation is 
now that if z { is any complex line in p, there is a pencil of lines in 
p with vertex on y t . Consequently in a singular plane the complex 
conic splits up into two pencils to which the singular line is common. 

We shall now show that any point at which the complex cone 
splits into two pencils is a singular point and any plane in which 
the complex conic splits into two pencils is a singular plane. 

Let A be such a point, and let the two pencils be a.+ \b { arid 
a i -f- fjLSf. Then 

2^;= o, 2)<?M= > 5) w<= - ( 10 > 

The tangent complex at a i contains a t ., 6 { , and e { by (10). There 
fore, by theorem V, 133, it is special, and the point A lies on its 
axis. Hence A is a singular point. The second part of the theorem 
is similarly proved. 

Now let a. and b t be two intersecting complex lines. Then 



If the pencil a i + \b t belongs entirely to the complex we have also 

V c.ab.= 0. (12^ 

/ ill XX 

We shall fix a i and take as b { that line of the pencil which 
intersects a fixed line d { which does not intersect a t . 

-0, 2>A*0. (13) 

To determine b i we have five equations of which 
three are linear and two quadratic. There are there 
fore in general four sets of values of b { , so that on 
any line of the complex there are in general four 
singular points. 

Let the four points be A^ A^ A^ A^ (Fig. 59) and 
the four lines be & , V, b" 1 , b"". Then each of the 
planes (& ), ("), (& "), (ab"") contains a pencil 
of lines and hence a second one distinct or coincident. 
Therefore through any line on the complex there are four singular planes. 

Since the coordinates of the four lines b i satisfy three linear 
equations, the lines belong in general to a regulus ( 140) and do 




LINE COORDINATES 331 

not intersect. Therefore the four points A are in general distinct, 
as are the four planes (#6). In order that two points or planes 
should coincide it is necessary that the regulus should degenerate, 
as in Case II, 140. The condition for this is that the discriminant 
of the equation 

X 2 5>?+ rt^ti+^W + 2 \r%aA+2 ^2> A 4+2 v^crf = 

should vanish. By virtue of (11), and the fact that d { satisfies (2), 
the above equation reduces to 



= ; 
and the condition that its discriminant should vanish is 



since ]a t 4^ 0, by (13). 

If this condition is met, a i is a singular line by the previous 
definition, two of the points A^ A^ A^ A^ coincide into one sin 
gular point on ., and two of the singular planes coincide. More 
precisely, if A l and A 2 coincide at A the pencils (> ) and (ab") 
form the complex cone at A, the two lines b fff and b"" intersect on d 
(compare 140), and the points A s and A 4 are the vertices of the 
pencils of complex lines in the plane (ab 1 " I""). 

142. Singular surface of the quadratic complex. The singular 
points and planes are determined by the complex line y { and the 
intersecting line c { y { , where ^ 6 ^~ ^ 

We take the pencil 



Then z t satisfies the equations 



or, what amounts to the same thing, the equations 

v z\ = o, V - 2 2? = o. (i) 

Equation (1) shows that z i is a singular line of the complex 



332 FOUR-DIMENSIONAL GEOMETRY 

Since the lines z i and belong to the same pencil as y. 

.+ X 

and cgji, the singular points and planes of (2) are the same as 
those of **cpl = 0, no matter what the value of X. The com 
plexes (2) are called cosingular complexes. 

We may use the cosingular complexes to prove that on any line 
in space lie four singular points of the complex .cjx% = 0, and through 
any line go four singular planes. 

Let I be any line in space. We may determine X in (2) so 
that / lies in the complex (2); in fact, this may be done in four 
ways, since (2) is of the fourth order in X by virtue of the relation 
V# t 2 = 0. Then there will be four singular points of this new com 
plex on I by previous proof, and these points are the same as the 
singular points of ^?CJK? = 0. 

It follows at once that the locus of the singular points of a quad- 
ratic complex ^ c i x f = * * a surface of the fourth order, and the 
envelope of the singular planes is a surface of the fourth class. 

These two surfaces, however, are the same surface. For if two of 
the singular points on I coincide, two of the singular planes through 
I also coincide. Therefore, if I is tangent to one of the surfaces it 
is tangent to the other. But I is any line. Therefore the two sur 
faces have the same tangent lines and therefore coincide. 

This surface, the locus of the singular points and the envelope 
of the singular planes, is called the singular surface. 

We shall not pursue further the study of the singular surface. 
Its Cartesian equation may be written down by first transform 
ing from Klein to P Kicker coordinates and replacing the latter 
by their values in the coordinates of two points (x, y, z) and 
(x , y , z ~). Then, if (x 1 , y , z ~) is constant, the equation is that 
of the complex cone through (x , y , z ). The condition that this 
cone should degenerate into a pair of planes is the Cartesian equa 
tion of the singular surface. It may be shown that the surface 
has sixteen double points and sixteen double tangent planes 
and is therefore identical with the interesting surface known as 
Rummer s surface.* 

* Cf. Salmon-Rogers, "Analytic Geometry of Three Dimensions," and Hudson, 
" Rummer s Quartic Surface." The latter book contains as frontispiece a photo 
graph of the surface. 



LINE COORDINATES 333 

EXERCISES 

1. Prove that the tangent lines of a fixed quadric surface form a 
quadratic complex. Find the singular surface. Note the peculiarities 
when the quadric is a sphere. 

2. Prove that the lines which intersect the four faces of a fixed tet 
rahedron in points whose cross ratio is constant form a quadratic com 
plex- whose equation may be written Ap 12 p M + Bpi 3 P 42 + c PuPm - 
This is the tetrahedral complex. 

3. Prove that in a tetrahedral complex all lines through any vertex 
or lying in any plane of the fixed tetrahedron belong to the complex. 
Find the singular surface. 

4. Show that lines, each of which meets a pair of corresponding lines 
of two projective pencils, form a tetrahedral complex. 

5. Show that the lines connecting corresponding points of a collinea- 
tion form a tetrahedral complex. 

6. If the coordinates of two lines x t and y i are connected by the 
relations 

005,= 

show that Xi belongs to the complex *CJK* = and that y { belongs to 
the cosingular complex 



7. If x i and x\ are two lines of a complex C, and y { and y\ their 
corresponding lines, as in Ex. 6, of a cosingular complex C\, prove the 
following propositions : 

(1) If Xi intersects y\, then x\ intersects y L . 

(2) If Xi intersects x\ at P, and y i intersects y\ at Q, the complex 
cone of C at P and the complex cone of C A at Q degenerate into plane 
pencils, and to a pencil of either complex corresponds a pencil of 
the other. 

(3) If x { intersects x\ at P, in general y { does not intersect y\, and the 
complex cone of C at P corresponds to a regulus of C\. Also the com 
plex conic in the plane of x { and x\ corresponds to a regulus of C A . 

(4) Any two lines x i and x[ of C which do not intersect determine a 
cosingular complex C\ in which the two lines y { and y it corresponding 
to x { and x { , intersect. There are, therefore, two reguli of C through x { 
and x\ corresponding to the complex cone and the complex conic of C A 
determined by y { and y\. 



334 FOUR-DIMENSIONAL GEOMETRY 

8. Prove that for an algebraic complex f(x v oj a , jc 8 , 4 , x 5 , x 6 ) = of 
the degree n the singular lines are given by the equations 



and that the singular surface is of degree 2 n (n I) 2 , where singular 
line and surface are defined as for the quadratic complex. 

143. Pliicker s complex surfaces. In any arbitrarily assumed 
plane the lines which belong to a given quadratic complex envelop 
a conic. If the plane revolves about a fixed line, the conic describes 
a surface called by Pliicker a meridian surface of the complex. 
If the plane moves parallel to itself, the conic describes a sur 
face called by Pliicker an equatorial surface of the complex. It is 
obvious that an equatorial surface is only a particular case of 
the meridian surface arising when the line about which the plane 
revolves is at infinity. In either case the surface has been called a 
complex surface. 

It is not difficult to write down the equation of a complex sur 
face. Let the line about which the plane revolves be determined 
by two fixed points, A and B, let P be any point in space, and let 
u t and v t be the coordinates of the lines PA and BP respectively. 

Then the coordinates of any line of the pencil defined by PA 
and PB are u { + \v { , and this line will belong to the quadratic 
complex ^\c { xf = when X satisfies the equation 

, + 2 X W( + X 2> ( " = 0. (1) 

In general there are two roots of this equation, corresponding to 
the geometric fact that in any plane through a fixed point there 
are only two complex lines, the two tangents to the complex conic 
in that plane. If, however, P is on that conic, the roots of (1) 
must be equal ; that is 



Now u { involves the point coordinates of A and P linearly, and 
v { involves in a similar manner the coordinates of B and P. Hence 
(2) is of the fourth order in the point coordinates of P. 

From the construction P is any point on the complex surface 
formed by the revolving plane about the line AB. Hence Pliicker s 
complex surfaces are of the fourth order. 



LIKE COORDINATES 335 

We may work in the same way with plane coordinates ; that 
is, we may define a straight line by the intersection of two fixed 
planes, a and /3, and take M as any plane in space. Then the three 
planes fix a point on , and equation (1) determines the two lines 
through that point in the plane M which belong to the quadratic 
complex. Hence, if the coordinates of M satisfy equation (2), M is 
tangent to the complex cone through that point on I. A little 
reflection shows that such a plane is tangent to the complex sur 
face formed by revolving a plane about the line I and that any 
tangent plane to the complex surface is tangent to a cone of com 
plex lines with its vertex on I. Hence (2) is the equation in plane 
coordinates of the complex surface. Therefore a complex surface 
is of the fourth class. 

144. The (2, 2) congruence. Consider the congruence defined by 
the two equations 

* = 0, (1) 

2>,^=0, (2) 

which consists of lines common to a linear and a quadratic 
complex. Through every point of space go two lines of the con 
gruence ; namely, those common to the pencil of lines of (1) and 
the complex cone of (3) through that point. Similarly, in every 
plane lie two congruence lines which are common to the pencil 
of (1) and the conic of (2) in that plane. The complex is there 
fore of second order and second class and is called the (2, 2) 
congruence. 

Consider any line y { of the congruence, and P any point on it. 
Through P there will go in an exceptional manner only one con 
gruence line, when the polar plane of P with respect to (1) coincides 
with the polar plane of P with respect to the tangent linear com 
plex of (2) at y.. This will occur at two points on y.. This may 
be seen without analysis from the fact that to every point on y i 
may be associated two planes through y i ; namely, the polar planes 
with respect to (1) and to the tangent linear complex at y.. Hence 
these planes are in a one-to-one correspondence, and there are two 
fixed points of such a correspondence. 

Analytically, if the complex (1) and the tangent linear complex 
of (2) have at P any line z,. in common distinct from y., they will 



336 FOUR-DIMENSIONAL GEOMETRY 

have the entire pencil y { + \z { in common. The conditions for 
this are 



This determines a line series which, by 140, degenerates into 
two plane pencils with vertices on y.. 

The points on y i with the properties just described are called 
the focal points F^ and F Z of y { , and the planes of the common 
pencil of (1) and the tangent linear complex of (3) are called 
the focal planes f^ and / 2 . The focal points are often described 
as the points in which y. is intersected by a consecutive line. The 
meaning of this is evident from our discussion. For at F l and F Z 
the pencil of lines of (1) is tangent to the complex cone of (2), so 
that through F l or F z goes only one line of the congruence doubly 
reckoned. 

The locus of the focal points is the focal surface. It will be 
shown in the next section that the line y i is tangent to the focal 
surface at each of the points F l and F^ and that the planes f^ and 
/ are tangent to the same surface at F Z and F l respectively. 

145. Line congruences in general. A congruence of lines consists 
of lines whose coordinates are functions of two independent vari 
ables. For convenience we will return to the coordinates first 
mentioned in 127 and, writing the equation of a line in the form 

x = rz 4- , y = pz + (r, (1) 

will take r, s, /a, and a- as the coordinates of the line. Then, if 
r, , /3, a- are functions of two independent variables a, /3, the lines 
(1) form a congruence. 

Let I be a line of the congruence for which a = , fi = f$ Q . If we 



we arrange the lines into ruled surfaces ; and if we further impose 
on </>() the single condition 



LINE COORDINATES 337 

we shall have all ruled surfaces which are formed of lines of the 
congruence and which pass through I. 

It is desired to know how many of these surfaces are develop- 
ables. For this it is necessary and sufficient that there exist a 
curve C to which each of the lines of the surface are tangent. The 
lines of the surface being determined by (1), (2), and (3), the 
coordinates of C are functions of a. The direction dx: dy. dz of C 
therefore satisfies the equations 

dx = rdz -\-zdr-\- ds, 
dy p dz + zdp -f- dcr, 

(dr dr \ 

4- < (#) ) da, and similar expressions hold for 
dec dp ) 

ds, dp, dcr. On the other hand, the direction of the straight line (1) 

is given by , 

dx = rdz, dy = p dz, 

so that if the straight line and curve are tangent, z must satisfy 
the two equations 

zdr + ds = 0, zdp + do- = 0, 
and therefore we must have . 

dpds drdcr 0. 

If we replace dr, ds, dp, dor by their values, we have as an equation 
for < one which can be reduced to the form 



Aft 2 (a) + J5< (a) + <7 = 0. 

From this equation with the initial conditions (3) we determine 
two functions </>(). They have been obtained as necessary con 
ditions for the existence of the developable surface through I, but 
it is not difficult to show that if <() is thus determined, the devel 
opable surface really exists. Hence we have the theorem : 

Through any line of a congruence go two developable surfaces 
formed by lines of the congruences. 

Of course it is not impossible that the two surfaces should coin 
cide, but in general they will not, and we shall continue to discuss 
the general case. 

To the two developable surfaces through I belong two curves 
C l and C z , the cuspidal edges to which the congruence lines are 



338 FOUR-DIMENSIONAL GEOMETRY 

tangent. The points F l and F^ at which I is tangent to C l and C 2 , 
are the focal points on I. The locus of the focal points is the focal 
surface. 

It is obvious that any line of the congruence is tangent to the 
focal surface, for it is tangent to the cuspidal edge of the devel 
opable to which it belongs, and the cuspidal edge lies on the 
focal surface. 

Let the line I be tangent to the focal surface at F l and F^ and 
let Cj be the cuspidal edge to which I is tangent at F^ Displace I 
slightly along C l into the position I tangent to C 1 at F[. The line 
I is tangent to the focal surface again at F, and the line F Z F! Z is 
a chord of the focal surface. As the point F[ approaches F l along 
CP the chord F^F^ approaches a tangent to the focal surface at F 21 
and the plane of I and V therefore approaches a tangent plane to 
the focal surface at F z . But this plane is also the osculating plane 
of the curve (7 1 . Hence the osculating plane of the curve C l at F I is 
tangent to the focal surface at F^. 

An interesting and important example of a line congruence is 
found in the normal lines to any surface, for the normal is fully 
determined by the two variables which fix a point of the surface. 
Through any normal go two developable surfaces which cut out 
on the given surface two curves which are called lines of curvature. 
These curves may also be defined as curves such that normals to 
the given surfaces at two consecutive points intersect, for this is 
only one way of saying that the normals form a developable 
surface. Through any point of the surface go then two lines of 
curvature. 

The two focal points on any normal are the centers of curvature. 
The distance from the focal points to the surface are the principal 
radii of curvature, and the focal surface is the surface of centers 
of curvature. The study of these properties belongs properly to 
the branch of geometry called differential geometry and lies out 
side the plan of this book. We will mention without proof the 
important theorem that the lines of curvature are orthogonal. 

We shall, however, find room for one more theorem ; namely, 
that a congruence of lines normal to one surface is normal to the 
family of surfaces which cut off equal distances on every normal 
measured from points of the first surface. 



LINE COORDINATES 339 

Let us write the equations of the normal in the form 

x = a -f Zr, 
y = P + mr, (4) 

z = 7 + nr, 

where (#, /3, 7) is a point of a surface S; l,m,n the direction cosines 
of the normal to S; and r the distance from S to a point P of the 
normal. Then 



whence Zt?/ -f- ?ft dm + n^w = 0. 

We have also Ida -f- widft + ndy = 0, 

since the line is normal to S. 

Suppose, now, we displace the normal slightly, but hold r constant. 
The point P goes into the point (x -f- dx, y + dy, z -f dz), where, 

from (4), 

dx = da + rdl, 

dy = dp + rdm, 
dz dj -f rdn ; 
whence Idx -\- mdy + ndz 0. 

That is, the displacement of P takes place in a direction normal 
to the line (4). From this it follows that the locus of points at a 
normal distance r from $ is another surface cutting each normal 
orthogonally, which is the theorem to be proved. 

EXERCISES 

1. Show that the focal points upon a line I of a congruence can be 
denned as the points at which all ruled surfaces which pass through /, 
and are composed of lines of the congruence, are tangent. 

2. Show that the singular lines of a quadratic complex form a con 
gruence, and that the singular surface of the complex is one nappe of 
the focal surface of the congruence. 

3. Show that in general there does not exist a surface normal to the 
lines of a congruence, and that the necessary and sufficient condition 
that such a surface exists is that the two developable surfaces through 
any line of the congruence are orthogonal. 



340 FOUR-DIMENSIONAL GEOMETRY 

4. Show that if a ruled surface is composed of lines of a linear 
complex, on any line of the surface there are two points at which the 
tangent plane of the surface is the polar plane of the complex. 

5. Consider any congruence of curves defined by 



/ 2 (jc, y, z, a, &)=0, 

and define as surfaces of the congruence surfaces formed by collecting 
the congruence curves into surfaces according to any law. Show that 
on any congruence curve C there exists a certain number of focal points 
such that all surfaces of the congruence which contain C are tangent 
at these points. 

6. Prove that if the curves in Ex. 5 are so assembled as to have an 
envelope, the envelope is composed of focal points. 



CHAPTER XVIII 

SPHERE COORDINATES 

146. Elementary sphere coordinates. Another simple example of 
a geometric figure determined by four parameters is the sphere. 
We may take the quantities d, e, f, r, which fix the center and 
radius of the sphere 

(x-d)*+(jf- e y+(z-f?=s, (i) 

as the coordinates of the sphere, and obtain a four-dimensional 
geometry in which the sphere is the element. 

It is more convenient, however, to use the pentaspherical coor 
dinates x i of a point and take the ratios of the coefficients . in 
the equation A 

Vi + Va + Vs + V* + V* = ( 2 ) 

of a sphere as the sphere coordinates. This is essentially the same 
as taking c?, e, f, and r. In fact, if x i are the coordinates of 117, 
then by (4), 117, equation (2) can be written 

Oi + K) ^ + f + * 2 ) + 2 V + 2^+2 a 4 z - <X - ifl 6 ) =0, (3) 

and the connection with (1) is obvious. 

By 119 two spheres are orthogonal when and only when 



the coordinates a? f being assumed orthogonal. 
Consider now any linear equation 



where c i are constants and w t . sphere coordinates. If we determine 
a sphere with coordinates c t ., (5) is the same as (4). Hence 

A linear equation in elementary sphere coordinates represents a 
complex of spheres consisting of spheres orthogonal to a fixed sphere. 
If the fixed sphere is special the complex consists of spheres through 
the center of the special sphere and is called a special complex. 

341 



342 FOUR-DIMENSIONAL GEOMETRY 

The word " complex " is used in the same sense as in 113, for 
if a., {$ 7 t ., 5 t . are four spheres which satisfy (4), any sphere which 
satisfies (4) has the coordinates 



Consider now the two simultaneous equations in sphere 
coordinates : ^c { u t = 0, ]^ = 0. (6) 

Spheres which satisfy both of these equations belong to two 
complexes. Therefore two simultaneous linear equations in elemen 
tary sphere coordinates are satisfied by spheres which are orthogonal 
to two fixed spheres. These spheres form a bundle, for if ., /:?., 7. 
are any three spheres which satisfy (6), any sphere satisfying .(6) 
has the coordinates a^-f- X/3.+ /j,y t . 

All spheres which belong to the two complexes in (6) belong 
to the complex ^0^+ X^t? j .w.= 0, and any two complexes of the 
latter form determine the bundle. Among these complexes there 
are in general two and only two special ones, and so we reach 
again the conclusion that a bundle of spheres consists in general 
of spheres through two fixed points. 

Three linear equations, 

]T C.M . = 0, ^ d t u t = 0, 2) e ^ u i = 

determine spheres which are orthogonal to three base spheres. 
These spheres form a pencil, since if a. and /3. are any two spheres 
satisfying (7), any sphere which satisfies (7) has the coordinates 
ffi+Xft 

We shall not proceed further with the study of the elementary 
coordinates, as more interest attaches to the higher coordinates, 
defined in the next section. 

EXERCISES 

1. Consider the quadratic complex ^a ik UfU k = 0, (a ki = a ik ) and 
the polar linear complex of a sphere v i} defined by the equation 
5} a a*Wfc = 0- If the determinant a ik 3= 0, show that to any sphere v t 
corresponds one polar complex, and conversely. 

2. Show that if v { lies in the polar complex of w iy then w t lies in 
the polar complex of v { . The two spheres v t and w t are said to be 
conjugate. 



SPHERE COORDINATES 343 

3. Show that the pencil of spheres denned by two conjugate spheres 
has in common with the quadratic complex two spheres which are 
harmonic conjugates of the first two spheres (the cross ratio of four 
spheres of a pencil is defined as in the case of pencils of planes). 

4. Show that the assemblage of all special spheres forms a quadratic 
complex. Show that any two orthogonal spheres are conjugate with 
respect to this complex, and that the polar complex of any sphere v t is 
the complex of spheres orthogonal to v,.. 

5. Show that the planes which belong to a quadratic complex en 
velop a quadric surface. 

6. Show that any arbitrary pencil of spheres contains two spheres 
which belong to a given quadratic complex, and that any arbitrary point 
is the center of two spheres of the complex. 

7. Show that the locus of the centers of the point spheres of a 
complex with nonvanishing discriminant is a cyclide. 

8. Define as a simjily special complex one for which the discriminant 
\a it . vanishes but so that all its first minors do not vanish. Show that 

| IK 

such a complex contains one singular sphere which is conjugate to all 
spheres in space. Show that the complex contains all spheres of the 
pencil determined by the singular sphere and any other sphere of 
the complex, and that all spheres of such a pencil have the same polar 
complex. 

147. Higher sphere coordinates. Let x i be orthogonal penta- 
spherical coordinates whereby 



and let a i^i+ a &+ Vs + a p<+ a & X 5 = 

be the equation of a sphere. To the five quantities 1? 2 , # 8 , 4 , 
we will adjoin a sixth one, # 6 , defined by the relation 



a*+a*+al. (3) 

The six quantities are then bound by the quadratic relation 

f (a)= a *+ a*+ 3 2 + <+ a 5 2 + = 0, (4) 

and the ratios of these quantities are taken as the coordinates of the 
sphere. This is justified by the fact that if the sphere is given, 
the coordinates are determined ; and if the coordinates are given, 
the sphere is determined. 



344 FOUK-DIMENSIONAL GEOMETRY 

More generally, if a^ # 2 , # 3 , a^ # 5 , a & are six quantities such that 



with the condition that the determinant tfj shall not vanish, the 
ratios a i : a k may be used as the coordinates of the sphere. Equa 
tion (4) then goes into a more general quadratic relation. We 
shall, however, confine ourselves to the simpler a,. 
By (20), 121, the radius of the sphere 



is 



Consequently, to change the sign of a & is to change the sign of the 
radius of the corresponding sphere. If, then, we desire to maintain 
a one-to-one relation between a sphere and its coordinates, we must 
adopt some convention as to the meaning of a negative radius. 
This we shall do by considering a sphere with a positive radius as 
bounding that portion of space which contains its center, and a 
sphere with negative radius as bounding the exterior portion of 
space. Otherwise expressed, the positive radius goes with the inner 
surface of the sphere, the negative radius with the outer surface. 
A sphere with its radius thus determined is an oriented sphere. 

If the sphere becomes a plane the positive value of 6 is associ 
ated with one side of the plane, the negative value with the other. 

A sphere is special when and only when 6 = 0. 

148. Angle between spheres. By 119 the angle between two 
spheres with coordinates a. and b { is defined by the equation 



a A 

Hence the angle 6 is determined without ambiguity when the 
signs of the radii of the two spheres are known. If both radii are 
positive, 6 is the angle interior to both spheres; if both radii are 
negative, 6 is exterior to both spheres ; and if the radii are of opposite 
sign, 6 is interior to one sphere and exterior to the other. 

For special spheres the angle defined by (1) becomes indeter 
minate. More precisely, if a t is a special sphere the coordinate 



SPHERE COORDINATES 345 

# 6 = and the other five sphere coordinates are the pentaspherical 
coordinates of the center of the sphere. Therefore the condition 
that the center of the special sphere a { lie on another sphere b { is 



Therefore if a f is a special sphere, b { any other sphere, and 
the angle between a t . and 5 t ., cos is infinite when the center of 
a. t does not lie on 5., but is % when the center of a t . lies on I.. 
A special sphere therefore makes any angle with a sphere on which 
its center lies. 

When 8 = (2 k + 1) |, ij (a, 6) = ,*, + ,*, + a,J, + A + .i. = 0, 

and conversely. Hence we may say : 

The vanishing of the first polar of 77 (a) is the condition that two 
spheres be orthogonal. 

When (9=0, f (a, 5)= ^+ ^ 2 + a/ 3 + a 4 6 4 + 5 & 5 + 6 ^ 6 = 0, and 
conversely. In this case the spheres are said to be tangent, but it 
is to be noticed that spheres are not tangent when 6 = TT. The dif 
ference between the cases in which 6 = and those in which 6 = TT 
lies in the relation to each other of the space which the spheres 
bound. In fact, if two spheres which are tangent in the elementary 
sense lie outside of each other, they are tangent in the present 
sense only when one is the boundary of its interior space, and the 
other is the boundary of its exterior space ; that is, the two radii 
have opposite signs. If two elementary spheres are tangent so that 
one lies inside the other, they are tangent when oriented only if 
the radii have the same sign. We say: 

The vanishing of the first polar of % (a) is the condition that two 
spheres be tangent. 

Two planes are tangent when they are parallel or intersect in a 
minimum line (Ex. 8, 81). 

It is obvious that all these theorems are unaltered by the use 
of the more general sphere coordinates of 121. 

The angle O k made by the sphere a i with the coordinate sphere 
x k = is given by the equation 



346 FOUR-DIMENSIONAL GEOMETRY 

Consequently we have the theorem : 

By the use of orthogonal coordinates x { and the sphere coordinates a r , 
the five coordinates a^ a^ 3 , 4 , a 5 of any sphere are proportional to the 
cosines of the angles which that sphere makes with the coordinate 
spheres. 

149. The linear complex of oriented spheres. Equation (1) of 
148 may be written 

A + fl A+ a &+ a A + ^5+ a ^ cos e = - 
Consider now a linear equation 

C 1 U 1+ C M S + W.+ W+ C 6 M 6 + W = 

where w t . are higher sphere coordinates and c. are constants. The 
spheres which satisfy this equation form a linear complex. 

This equation may in general be identified with (1) by deter 
mining a fixed sphere, called the base sphere, with the coordinates 
.= <?., (i = l, 2, 3, 4, 5), 6 - ijc* + cl + c 3 2 + el + c 6 2 , (3) 
and determining an angle 6 by the equation 

a & cos 6 = c & . (4) 

Equation (2) is then satisfied by all spheres which make the 
angle 6 with the base sphere. This angle is equal to when and 
only when c & a & ; that is, when f (<?) = 0. In the latter case the 
complex is called special. 

We put these results in the form of the theorem : 

A linear complex consists in general of spheres cutting a fixed 
sphere under a constant angle. If % (c) = the complex is special 
and consists of spheres tangent to a fixed sphere. 

The words "in general" have been introduced into the theorem 
because of the exceptional cases which arise when the base sphere 
is special ; that is, when a & = 0. In that case the angle 6 cannot be - 
determined from (4). 

If at the same time that 6 = the complex is special, then 6 = 0, 
and the complex is 



with ^cf= 0. Then c. are the coordinates of a point, the center of 
the base sphere, and hence a special complex may consist of spheres 
intersecting in a point. 



SPHERE COORDINATES 347 

If when a & = the complex is not special, then c 9 =f= 0, and the 
angle 6 cannot be determined. A particular case in which this may 
happen is when c l = c z = c s = c^ c 5 = 0, and the complex is 

6 =o. 

This equation is satisfied by all special spheres. Therefore all 
special spheres together form a nonspecial linear complex in which the 
base sphere is indeterminate. 

There remain still other cases in which a = 0, but c a = 0. The 

6 6 

base sphere is then special and the angle 6 is infinite, but the com 
plete definition of the complex is through its equation. 

EXERCISES 

1. Prove that the base sphere of a complex is the locus of the 
centers of the special spheres which belong to the complex. 

2. Prove that if c 6 = in the equation of a complex, the complex 
consists of spheres orthogonal to a fixed sphere, as in 146. 

3. Prove that in a special complex the coefficients in the equation 
of the complex are the coordinates of the base sphere. 

4. Prove that all planes together make a special complex with the 
base sphere the locus at infinity. 

5. Show that all spheres with a fixed radius form a linear complex 
and determine the base sphere. 

6. Discuss the relation between two complexes whose equations 
differ only in the sign of the last term. 

7. Two linear complexes ^c f u { = and ^d,n,= being said to be 
in involution when c^ + c 2 d 2 -f- c^d s -f- c^d 4 -f- c 5 c? 5 + c & d 6 = 0, show that 
when the base spheres of the two complexes are nonspecial, the product 
of the cosines of the angles which the spheres of each complex makes 
with its base sphere is equal to the cosine of the angle between the 
base spheres. 

8. Prove that a special complex is in involution with every complex 
which contains its base sphere. 

9. Show that the complex consisting of spheres orthogonal to a 
nonspecial base sphere is in involution with the complex of all special 
spheres. 

10. Show that the six complexes u { are pair by pair in involution 
and determine the relations of the base spheres. 



348 FOUR-DIMENSIONAL GEOMETRY 

11. Conjugate spheres with respect to a linear complex are such that 
any sphere tangent to both belongs to the complex, and any sphere of 
the complex tangent to one is tangent to the other. 

Show that if v t is any sphere, the conjugate sphere has tke coordinates 



12. If a complex is composed of spheres orthogonal to a base sphere, 
show that the conjugate of a sphere S is the inverse of S with respect 
to the base sphere. 

13. Find without calculation and verify by the formulas the con 
jugate of a sphere with reference to a complex of spheres with fixed 
radius R. 

14. Show that the conjugate of a sphere with respect to the complex 
of special spheres is the same sphere with the sign of the radius changed. 

150. Linear congruence of oriented spheres. The spheres common 
to two linear complexes 



form a sphere congruence. Any sphere of the congruence (1) also 
belongs to any complex of the form 



=0, (2) 

and any two complexes of form (2) can be used to define the 
congruence. 

Now (2) represents a special complex when X satisfies the 
equation 



that is, f (a) + 2 X (a, ft) + X 2 f (6) = 0. (3) 



Hence, in general, a sphere congruence consists of spheres tangent 
to two spheres, called directrix spheres. 

The exceptional cases occur when the roots of equation (3) 
are either illusive or equal. In the first case equation (3) is 
identically satisfied and all complexes of (2) are special. The 
congruence may then be defined in an infinite number of ways 
as composed of spheres tangent to two directrix spheres. The 
condition that (3) be identically satisfied is f(a) = 0, 



SPHERE COORDINATES 349 

f (a, >) = 0. The first two equations say that the defining com 
plexes are special; the third equation says that the base sphere 
of either lies on the other. 

If the two roots of (3) are equal, there is only one special com 
plex in the pencil (2). Suppose we take this as 2 a < M < !Ba ^- Then, 
since the roots of (3) are equal, f (a, ft) = 0. This says that 
the base sphere of the special complex belongs to the complex 

S,u,= o. 

151. Linear series of oriented spheres. Consider now the spheres 
common to the three complexes 



which do not define the same congruence. These spheres form a 
linear series. 

A sphere of the series (1) belongs also to any complex of the 

f rm 



-^+i^t^O, (2) 

and any three linearly independent complexes (2). may be used to 
define the series. Among the complexes (2) there are a simply 
infinite set of special complexes ; namely, those for which X, /*, and 
v satisfy the equation f (Xa + ^j + , ) = . (3) 

The spheres of the series (1) form, therefore, a one-dimensional 
extent of spheres which are tangent to a one-dimensional extent of 
directrix spheres. 

The nature of the series depends on the character of equation (3). 

We shall assume that the discriminant of (3) does not vanish. 
If the quantities (X, ft, v) are for a moment interpreted as trilinear 
point coordinates in a plane, equation (3) will represent a conic 
without singular points ; hence it is possible to find three sets of 
values which satisfy (3) and- are linearly independent. We have 
corresponding to these values of (\, p, v) three linearly independent 
special complexes, and may assume without loss of generality that 
they are the three complexes in equations (1). 

Then any one of the directrix spheres has the coordinates 

(149) /^ = Xa t .+ /^ t .+ ^, (4) 

where (X + /*6 + )= 0, (a)=0, f(&)=0, fO) = 0. (5) 



350 FOUR-DIMENSIONAL GEOMETRY 

Now if atp /3 t ., and y i are any three spheres of the series (1), it is 
obvious that the spheres v { in (4) satisfy the three equations 

2/w = o, 5X*,- = o, 2)w = o. (6) 

Conversely, any sphere satisfying equations (6) satisfy (4), for 
three solutions of (6) are a,., b { , c { , and the most general solution 
is therefore \a i + ph. + vc { , where (since v i are sphere coordinates) 
equation (3) must be satisfied. 

Hence the directrix spheres form another linear series. 

The special complexes which may define the series (6) are 



where f (pa. + <r. + ry t .) = 0. 

The base spheres of these are simply the solutions of (1). Hence 
the directrix spheres of the series (6) are the spheres of (1). 

We have, therefore, two series of spheres such that each sphere of 
one series is the tangent to each sphere of the other. 

On the other hand, no two spheres of the same series are tangent. 
To prove this note that by (5) we have 

X/*f (a, b) + pvg (b, c) + i/Xf 0, a) = 0, 

and no one of these coefficients can vanish under the hypothesis 
that the discriminant of (3) does not vanish. But a fl b { , c l are any 
three directrix spheres, and hence the theorem. 
By 115 we are able to say immediately: 

In the general case the spheres of a linear series envelop a Lupin s 
cyclide. 

We shall not discuss the special forms of the linear series arising 
when the discriminant of equation (3) vanishes. 

152. Pencils and bundles of tangent spheres. If a. and b { are 

any two spheres, then . s 7 ,i\ 

fw,= a, + X6. (1) 

is a sphere when and only when V i 6 i = 0; that is, when a { and 
b. are tangent. In this case (1) represents oo 1 spheres, each of 
which is tangent to each of the others. We call this a pencil 
of tangent spheres. In the notation of 117 the condition for a 
special sphere in the pencil is 

6 +X6 6 =0, (2) 



SPHERE COORDINATES 351 

so that there is only one special sphere in the pencil unless a i and 
>;, and consequently all spheres of the pencil, are special. 
The condition for a plane in the pencil is 

a 1 + 2a 6 +\(J 1 +0 6 )=0, (3) 

so that there is only one plane in the pencil unless all the spheres 
of the pencil, including a i and 5 t , are planes. 

In general the special sphere and the plane are distinct from 
each other. Therefore the special sphere is a point sphere whose 
center is in finite space. This center lies on all spheres of the 
pencil by 148. Hence the pencil is composed of spheres tangent 
to each other at the same point. Such spheres have in common 
two minimum lines determined by the intersection of the point 
sphere and the plane of the pencil. These statements may be veri 
fied analytically by writing the equations of the spheres in the 
form (3), 111. 

Special forms of a tangent pencil may arise, however. For 
example, it may consist of spheres having two parallel minimum 
lines in common. The special sphere and the plane in the pencil 
then coincide with the minimum plane determined by these mini 
mum lines. Again, the pencil may consist of point spheres whose 
centers lie on a minimum line. The plane in the pencil is then 
the minimum plane through that line. Or the pencil may consist of 
parallel planes ( 48). The special sphere in the pencil is then the 
plane at infinity unless all the planes of the pencil are minimum 
planes and therefore special spheres. Finally, the pencil may 
consist of planes intersecting in the same minimum line ( 48). 
The special sphere is then the minimum plane through that line. 

If ., &., and c t are three spheres not in the same pencil, then 
jm^^+Xfi,. + /*<?,. (4) 

is a sphere when and only when the three spheres are tangent each 
to each. In that case equation (4) defines oo 2 spheres, each of which 
is tangent to each of the others. It is a bundle of tangent spheres. 
There are in the bundle oo l special spheres determined by the equation 

Vt-Xfl 6 +fM? 6 =0, (5) 

and oo * planes determined by the equation 
! 4- ia 6 



352 FOUR-DIMENSIONAL GEOMETRY 

In general, equations (5) and (6) have only one common solution, 
so that the special spheres are point spheres. Since all spheres of 
the bundle are tangent, the centers of the point spheres lie on a 
minimum line which lies on all the spheres of the bundle. The point 
spheres and the planes form each a pencil in the sense already dis 
cussed, so that any point of the common minimum line is the center 
of a point sphere of the bundle, and any plane through the minimum 
line is a plane of the bundle. From that we may show that any 
sphere which contains that minimum line and is properly oriented 
belongs to the bundle. For let v t be such a sphere and a[ any plane 
of the bundle. Since v. and a( have one minimum line in common, 
they have another minimum line in common which intersects the 
first one at a point P. Let b { be the point sphere with center P. 
Then v { is tangent both to a[ and b f { at P, and therefore 

pv t = a\ -f- r5J 

if the proper sign is given to a 6 . But a[= a t + V& i +/* c < and 
V i =a i +\"b i +iJk"c { , so that 



whence v t belongs to the bundle. 

Summing up, we say : In general a bundle of tangent spheres con 
sists of all the oo 2 spheres which have a minimum line in common 
and of no other spheres. 

To avoid misunderstanding the student should remember that 
we are dealing with oriented spheres and that, for example, three 
elementary tangent spheres which lie so that two of them are tan 
gent internally to the third, but externally to each other, cannot 
be so oriented as to be tangent in the sense in which we now use 
the word. 

Special forms of bundles deserve some mention. In the first 
place, we notice that not all the spheres can be point spheres ; since, 
if they were, the centers of three spheres would be finite points 
not in the same line but in the same plane, so that each is con 
nected with the other by a minimum line, which is impossible. 

The spheres of the bundle may, however, all be planes. Then 
the special spheres must be minimum planes, which, since they are 
tangent, must form a pencil of minimum planes tangent to the 
circle at infinity at the same point ( 48). All planes of the bundle 



SPHERE COORDINATES 353 

must pass through this point, and it is evident that any two 
planes through this point either intersect in a minimum line or are 
parallel, and in each case are tangent. Hence, as a special case a 
bundle of tangent spheres may consist of oo 2 planes through the same 
point on the imaginary circle at infinity. 

153. Quadratic complex of oriented spheres. Consider the quad 
ratic complex denned by the equation 

5>,?= o- (i) 

This is the form to which in general an equation of the second 
degree in x { can be reduced, and we shall consider only this case. 
Since the sphere coordinates satisfy the equation 



the same complex (1) is represented by any equation of the form 



Now let y i be a sphere of (3), and z. any sphere tangent to y 
and consider the pencil of tangent spheres 

puyt+ter (4) 

This pencil has in common with (3) the two spheres corre 
sponding to the values of X obtained by substituting from (4) in (3). 
This gives, with reference to the fact that y { satisfies (3), 



The one common sphere is, then, always y t , as it should be, but 
the other is in general distinct from y. and coincides with it when 
and only when z l satisfies the relation 



that is, when e. lies on the linear complex 



This complex is called a tangent linear complex. 

From the derivation a tangent linear complex through a sphere y. 
is a linear complex which contains y. and has the property that any 
pencil of tangent spheres belonging to the linear complex which 



354 FOUK-DIMENSIONAL GEOMETRY 

contains y i has, in common with the quadratic complex, only the 
sphere y i doubly reckoned, unless the pencil lies entirely in the 
quadratic complex. 

This definition is analogous to that given in point space for a 
tangent plane to a surface by means of coincident points of inter 
section of a line in the tangent plane. The exceptional cases of 
pencils entirely on the complex are analogous to tangent lines 
which lie entirely on the surface. 

It may also be noted that if y t -\- dy i is any sphere of (1) adja 
cent to y t , so that ^$#{ly f *= and, from (2), yj%y t ~ 0, the sphere 
lies also in (5). The tangent linear complex contains all spheres 
of the quadratic complex adjacent to ?/.. 

Since //. is arbitrary in (5) the quadratic complex (1) has a pencil 
of tangent linear complexes through any sphere y.. Among these 
there is in general one and only one which is a special complex, 
for the condition that (5) be special is 



which, if we replace ^ by and use (1) and (2), becomes 



The special linear tangent complex is then in general (/^ 2 = 0) 

5^:= 0. 

In an exceptional manner, however, all tangent linear complexes 
are special when V^2 2=0. ^ 

When this condition is satisfied the sphere y i is called a singular 
sphere. 

The conditions to be satisfied by the coordinates of a singular 
sphere are, accordingly, 



which express respectively that y. satisfies the fundamental equa 
tion for sphere coordinates, that the sphere y. is in the complex (1), 
and that it is a singular sphere. 

The last equation also expresses the fact that c ( y { are the coor 
dinates of some sphere, and the second equation tells us that the 
sphere c^y i is tangent to the sphere y? The two spheres therefore 



SPHERE COOKDLNTATES 355 

define a pencil. On the sphere y i there is, therefore, a definite 
point P, the center of the point sphere of the pencil. The locus of 
P is an oo 2 extent of points forming the surface of singularities. 

In order to determine the degree of the surface of singularities 
we shall take 2-, any sphere of the pencil of tangent spheres defined 
by y t and <y, p so that ^_ (( , + x) ^ (g) 

Substitution in (7) gives the equations 

y 3 = V C{ ^ = V G ^ = Q 
*(a { +\y ^(^-+X) 2 A^.+ x) 2 

but simple linear combinations of these show that they are equiv 
alent to the three equations 

S* =o. 2:^=0, S^-o- oo 



Conversely, if z f is any solution of (9) and we place u { 






A. 

it is clear that u. is a singular sphere of the quadratic complex (1). 
Therefore equations (9) are satisfied by all spheres belonging to 
any pencil of tangent spheres defined by a singular sphere y i and 
the sphere cy., and, conversely, any sphere which satisfies (9) 
belongs to such a pencil. 

Let us now adjoin the condition that z i should be a point sphere ; 
namely, ^ =a (1Q) 

Equations (9) and (10), then, define the points P. 
Consider now any straight line I defined as the intersections of 
two planes M and N. Take 

2>A= (11) 

as the equation of any linear complex which has M as a base 
S P here and 2> A .= (12) 

as the equation of any linear complex which has N&s a base sphere. 

The point spheres of the complex (11) have centers on M, and 
the point spheres of the complex (12) have centers on N, so that 
the point spheres belonging to M and N have centers on the line I. 

Hence the simultaneous solutions of equations (9), (10), (11), 
and (12) give the point spheres whose centers lie both on the 
surface of singularities and on the line I. The number of these 



356 FOUR-DIMENSIONAL GEOMETRY 

solutions is the number of points in which I meets the surface of 
singularities; that is, the degree of the surface. 

To solve these equations we may begin by eliminating X from 
the last two of equations (9). Since the third equation of (9) is 
the derivative of the second with respect to X, the elimination of 
X gives the condition that the second equation should have equal 
roots in X. Since the second equation in (9) is of the fourth order 
in X, by virtue of the first equation in (9), the result of the elim 
ination of X is an equation of the sixth degree in zf or the twelfth 
degree in 2 t , This equation, combined with the first of equa 
tions (9) and the linear equations (10), (11), (12), gives twenty- 
four solutions. Therefore the equation of singularities is of the 
twenty-fourth order. 

Equations (9)-(12) may be otherwise interpreted by consider 
ing (11) and (12) as the equations of two complexes with base 
spheres which are not planar and therefore intersect in a circle, 
which may be any circle. The special spheres of the complexes 
have their centers on this circle, and the special spheres which also 
satisfy (7)- (9) are point spheres, since the condition that they be 
planar adds a new equation which in general cannot be satisfied. 
Hence, by the argument above, any circle, as well as any straight 
line, meets the surface of singularities in twenty-four finite points. 

If the equations are expressed in Cartesian coordinates, the 
circle will meet a surface of the twenty -fourth order in forty-eight 
points. We have accounted for twenty-four finite points ; the other 
twenty-four must lie on the imaginary circle at infinity. Since the 
plane of the finite circle meets the circle at infinity in two points, 
we have the theorem : The surface of singularities contains the 
imaginary circle at infinity as a twelvefold line. 

Return, now, to the pencil (8). There is one plane p in the 
pencil which is tangent to y. at P and is uniquely determined by 
y { . Such planes form an oo 2 extent which envelop a surface. To 
show that this surface is the surface of singularities let y i H- cfo/ t be 
a singular sphere neighboring to y^ so that 



The pencil of tangent spheres defined by y i -\-dy i and c^y^dy^) is 

(14) 



SPHERE COORDINATES 357 

and the condition that v { should be tangent to 2 t is satisfied by 
virtue of (7) and (13). Hence, in particular, the point P, the center 
of the point sphere of (8), lies in the plane p of the pencil (14) ; 
that is, P is the limit point of intersection of two neighboring 
planes p and is therefore a point of the surface enveloped by p. 
This establishes the identity of the surface which is the locus of 
P and that enveloped by p. 

The class of the surface of singularities is the number of the 
planes p which pass through an arbitrary line. To determine this 
number we may again set up equations (9), (11), and (12), but 
replace (10) by ^ + ^ = 0> (15) 

which is the condition that u f should be a plane. 

Any plane of either of the c omplexes (11) or (12) intersects 
the base plane M or N respectively in a straight line, and therefore 
the planes common to M and N pass through the line /. The solu 
tions of equations (9), (11), (12), and (15) give, therefore, the 
planes tangent to the surface of singularities which pass through I. 
Hence the surface of singularities is of the twenty-fourth class. 

154. Duality of line and sphere geometry. Since line coordinates 
and higher sphere coordinates each consist of the ratios of six quan 
tities connected by a quadratic relation, there is duality between 
them. To bring out the dualistic properties we shall interpret the 
ratios of six quantities x { connected by the relation 

x$+ xl+xl+xl -f xl+ xl= 0, 

on the one hand, as the sphere coordinates a { of 147 and, on the 
other hand, as the Klein line coordinates of 130. 

It is to be noticed that for a real line, as shown in 130, we 
have x^ # 2 , # 4 real and o? 8 , o? 6 , x & pure imaginary. On the other 
hand it follows from 146, 147 that for a real sphere we have 

x i> x tf x & x rea l an ^ x tf X Q P ure i ma g mar y- Hence configurations 
which are real in either the line or the sphere space will be 
imaginary in the other. 

It is also to be noticed that a sphere for which # 6 = is peculiar, 
being a special sphere, but the line for which X G = has no special 
geometric properties. The complex of lines # 6 = has, however, 
a peculiar role in the dualistic relations. We shall call this com 
plex C. Its equation in Pliicker coordinates is p u P 2Z 0. 



358 



FOUR-DIMENSIONAL GEOMETRY 



Two spheres whose coordinates differ only in the sign of # 6 are 
the same in the elementary sense, but two lines whose coordinates 
differ in the same way are distinct and conjugate with respect to 
the complex C. The relation between sphere and line is therefore 
in one sense one-to-two, but becomes one-to-one by the convention 
of distinguishing between two spheres which differ in the sign of 
the radius. 

Any sphere for which x l + ix b = is a plane, but the correspond 
ing line has no special geometric property. The complex of lines 
2^4. ix & = 0, however, will have a peculiar r61e in the duality. We 
shall call this complex S. It is special and consists of lines inter 
secting the line with coordinates 1 : : : : i. Its equation in 
Pliicker coordinates is p s4 = 0. 

We have now as immediate consequences of our previous results 
the following dualistic relations: 



Line space 

A straight line. 

A line of the complex C. 

A line of the complex S. 

A line of C but not of S. 

A line of S but not of C. 

A line of C and of S. 

Two lines conjugate with respect 
to C. 

Two intersecting lines. 

A nonspecial complex. 

A special complex consisting of 
lines intersecting a fixed line. 

A linear congruence consisting 
of lines intersecting two lines. 

A linear series forming one set 
of generators of a quadric surface. 

A quadratic line complex with 
its singular surface. 



Sphere space 

A sphere. 

A special sphere. 

A plane. 

A point sphere. 

An ordinary plane. 

A minimum plane. 

Two spheres differing only in 
the sign of the radius. 

Two tangent spheres. 

A nonspecial complex. 

A special complex consisting of 
spheres tangent to a fixed sphere. 

A linear congruence consisting 
of spheres tangent to two spheres. 

A linear series forming one of 
the families of spheres which en 
velop a Dupin s cyclide. 

A quadratic sphere complex with 
its singular surface. 



A pencil of lines corresponds to a pencil of tangent spheres, 
and a bundle of lines to a bundle of tangent spheres. Consider a 
point P and the oo 2 lines through it. They correspond in general 



SPHEEE COORDINATES 359 

to a bundle of tangent spheres which have in common a minimum 
line p ( 152). It is therefore possible in this way to set up a 
correspondence of the line space and the sphere space by which 
any point of the line space corresponds to a minimum line of the 
sphere space. 

An exception occurs when the point P of the line space lies on 
the axis of the complex S. Then all lines through P belong to S, 
and the corresponding bundle of spheres consists of planes which 
have in common only a point on the imaginary circle at infinity. 

Consider two points P and Q connected by a line I correspond 
ing to a sphere s. P corresponds in the first place to a bundle of 
spheres containing s and therefore, in the second place, to a mini 
mum line p on s. Similarly, Q corresponds to a minimum line <?, 
.also on s. If p and q intersect in a finite point Jf, the point sphere 
with center M belongs to both the bundle of spheres containing p 
and that containing q. Therefore the line corresponding to this 
point sphere must pass through P and Q. Hence I, since it corre 
sponds to a point sphere, is in this case a line of the complex C. 

Conversely, if / is any line of the complex C the minimum lines 
corresponding to P and Q lie on a special sphere and intersect. 

Otherwise, if I is not a line of the complex C the minimum lines 
do not intersect in a finite point and hence are two generators of 
the same family on s. 

Consider now the line I conjugate to / with respect to the com 
plex C. The points of this line correspond to generators of the 
same sphere s. But points of I and I are connected by a line of (7, 
and therefore the generators given by I intersect those given by I. 
Therefore the generators given by points of I and I belong to 
different families. 

Consider now the lines of a plane. They form a bundle which 
corresponds to a bundle of tangent spheres. It is therefore possible 
to set up a correspondence of line space and sphere space by 
which a plane corresponds to a minimum line. We have nothing 
new, however, since the lines which lie in a plane are. conjugate 
with respect to C of the lines which pass through a point. In fact, 
if we keep to the correspondence of point and minimum line it is 
not difficult to show that the oo 2 points of a plane correspond to 
oo 2 minimum lines, which can be arranged in oo 1 spheres which have 



360 FOUR-DIMENSIONAL GEOMETRY 

a minimum line in common, so that in this way a plane corre 
sponds to a minimum line. 

We may exhibit these results in the following table: 

Line space Sphere space 

A point. A minimum line. 

The points of a general line I. One set of generators of a 

sphere s. 

The points of I conjugate to I The other set of generators of s. 
with respect to C. 

The points on a line of C but The minimum lines on a point 

not of S. sphere (the lines of a minimum 

cone). 

The points of a line of S but The two families of minimum 

not of C and the points of the lines of a plane, 
conjugate line with respect to C. 

The points of a line common to The single family of minimum 

C and S. lines on a minimum plane. 

Consider now any surface F in the line space. We may find a 
corresponding surface in the sphere space as follows. Let P be 
any point on F and consider the pencil of tangent lines to F at P. 
These lines if infinitesimal in length determine a surface element. 

Corresponding to the pencil of tangent lines there is in the 
sphere space a pencil of tangent spheres which determine a point 
P and a tangent plane ; that is, another surface element. It may 
be noticed that the point P is the center of the point sphere which 
corresponds to the line of the complex C in the pencil of lines 
which lie in the surface element of F. 

We have in this way associated to a surface element in the line 
space a surface element of the sphere space. When the surface 
elements in the line space are associated into a surface F, the sur 
face elements in the sphere space form another surface, F , which 
corresponds to F. 

To any tangent line of F at P corresponds a tangent sphere of 
F at P . It is known from surface theory that consecutive to P 
there are two points Q and R on F such that a tangent line at 
either coincides with a tangent line at P. The tangents PQ and 
PR are the principal tangents at P. If the directions of one of 



SPHERE COORDINATES 361 

these tangents is followed on the surface, we have a principal 
tangent line (or an asymptotic line) on F. 

Corresponding to this, there are in the sphere space two con 
secutive points Q and R on F such that a tangent sphere at 
either coincides with a tangent sphere at P . If one of the direc 
tions P Q or P R is followed on F , we have a line of curvature 
of F 1 . 

Therefore, in the correspondence before us principal tangent lines 
on a surface in the line space correspond to lines of curvature on the 
corresponding surface in the sphere space. 

EXERCISES 

1. Show that the relation between line space and sphere space may 
" be expressed by the equations 

Zz = Tx-(X- iY)t y 

(X + iY)z = Ty-Zt, 

where x\y\z\t are Cartesian point coordinates in the line space and 
X : Y : Z : T are similar coordinates in the sphere space. Verify all the 
results of the text. 

2. Trace the analogies between the four-dimensional sphere geom 
etry and the three-dimensional point geometry with pentaspherical 
coordinates. 



CHAPTER XIX 

FOUR-DIMENSIONAL POINT COORDINATES 

155. Definitions. We shall now develop the elements of a four- 
dimensional geometry in which the ideas and methods of the ele 
mentary three-dimensional point geometry are used and which 
stands in essentially the same relation to that geometry as that 
does to the geometry of the plane. 

We shall define as a point in a four-dimensional space any set 
of values of the four ratios x-. x: x : x.i x. of five variables. In a 

12345 

nonhomogeneous form the point is a set of values of the four 
variables (x, y, z, w). 

A straight line is defined as a one-dimensional extent determined 
by the equations 

px ( =y t +\z a (t = l, 2, 3, 4,5) (1) 

where y { and z i are two fixed points and X is an independent variable. 

A plane is defined as a two-dimensional extent determined by the 

equations ^ =yj+H+/Wp (, = 1, 2 , 8, 4, 6) (2) 

where x it y { , z { are three fixed points not on the same straight line 
and X, fji are independent variables. 

A hyperplane is defined as a three-dimensional extent determined 
by the equations 

px { = y, + Xz. + iL Wi + w*,., (t = 1, 2, 3, 4, 5) (3) 

where y# z t , w# u { are four fixed points not in the same plane and 
X, /x, v are independent variables. 

From these definitions follows at once the theorem : 

/. A straight line is completely and uniquely determined by any 
two of its points, a plane by any three of its points which are not 
collinear, and a hyperplane by any four of its points which are not 
coplanar. 

The forms of equations (1), (2), (3) show that if the fixed points 
are given, the corresponding locus is completely determined. The 

362 



POINT COORDINATES 363 

theorem asserts that any points on the locus which are the same 
in number and satisfy the same condition as the given points may 
be used to define the locus. We shall show this for the plane (2). 
Let Y. be a point defined by equations (2) when X = X x , p = /^ ; 

that is, let , r 

Yi=Vi+ *&+!*& (4) 

Equations (2) may then be written 



which are of the type px { = Y t + X z^-h n w^ (5) 

Then any point which can be obtained from (2) can also be 
obtained from (5), and conversely. 

The discussion, however, assumes that Y. is not on the same 
straight line through z. and w i ; for if it were, the coordinates of Y. 
would not be of the form (4). In fact, to obtain from (2) points 
on the line y { z { in the plane (2) it is necessary to replace X and //- 

by the fractional forms -, write the equation of the plane as 

v v 



and then place v = 0. 

We have shown that in equations (2) the point y. may be 
replaced by any point not on the same straight line with z. and w r 
In the same manner each of the other points may be replaced, and 
the theorem is proved for the plane. 

The student will have no difficulty in proving the theorem for 
straight line and hyperplane. 

Another immediate consequence of the definitions is the theorem : 

//. If two points lie in a plane, the line determined by them lies 
in the plane ; if three points lie in a hyperplane, the plane determined 
by them lies in the hyperplane. 

The proof is left to the student. 

If we eliminate p, X, /z, v from equations (3) we have the result 

2/ 2 Z 2 W 2 z 

2/3 *3 W B W 3 = C 6 ) 



364 FOUR-DIMENSIONAL GEOMETRY 

Hence : 

///. Any hyperplane may be represented by a linear equation in 
the coordinates x { . 

Conversely : 

IV. Any linear equation in x { represents a hyperplane. 

Let V ,*,.= 



be such an equation, and let ?/., 2., w t , u { be four points satisfying 
the equation but not on the same straight line. Then we have 



and by eliminating a. from these equations and (7) we have an 
equation of the form (6) and thence equations of form (3). 

If we eliminate /o, X, /t from equations (2) we have the two 
equations 



= 0, 



(8) 



That is : 



V. Any plane may be represented by two linear equations in the 
coordinates x { . 

Conversely : 

VI. Any two independent linear equations represent a plane. 

Let 2)^=0, 2)6,^=0 (9) 

be such equations. Since they are independent, at least one of the 



determinants 



is not zero. Let us assume that 



The two equations can then be solved for x and x 



=0. 

4 5 

and thus 

reduced to two of the type (9) with a b = and 5 4 = 0. If #., z., w i 
are three points satisfying the equations but not on the same 
straight line, we may then eliminate a. and b. and obtain equations 
of the form (8) and finally of the form (2). 

In the same manner we may easily prove : 

VII. Any straight line may be represented by three linear equations, 
and any three independent linear equations represent a straight line. 



POINT COORDINATES 365 

As a special case of theorem IV, any one of the five equations 
%.= represents a hyperplane. Consider in particular x & = 0. The 
points in this hyperplane have the coordinates x^. x z : x 3 : x 4 , as in 
projective three-dimensional space, and the definitions of straight 
line and plane are the usual definitions. The two equations which 
represent a plane consist of the equation x^= and any other linear 
equation. If, then, the equation x & = is assumed once for all, a 
plane is represented by a single equation. Similarly, a straight line 
in x b = is represented by two equations besides the equation x & = 0. 
Obviously the difference between the representations of a plane in 
three-dimensional and four-dimensional geometry is similar to that 
between the representations of a straight line in two-dimensional 
and three-dimensional geometry. 

Just as plane geometry is a section of space geometry, so space 
geometry is a section of four-dimensional geometry, the three- 
dimensional space being a hyperplane of the four-dimensional space. 

156. Intersections. We shall proceed to give theorems concerning 
the intersections of lines, planes, and hyperplaiie*s. In reading these 
it may be helpful for the student to bear in mind that within the 
same hyperplane these theorems are the same as those of the ordi 
nary three-dimensional geometry, but differences emerge as we 
consider figures in different hyperplanes. 

/. Two hyperplanes intersect in a plane. All hyperplanes through 
the same plane form a pencil, and any two of these hyperplanes may 
be used to define the plane. 

The first part of this theorem follows immediately from 
theorem VI, 155. For the latter part we notice that any hyper 
planes of the pencil ^a^-f X^fyz^ intersect in the plane 
determined by 2X# t -= and ^0^= 0. 

II. Three hyperplanes not in the same pencil intersect in a straight 
line. All hyperplanes through the same line form a bundle, and any 
three of them not in the same pencil determine the line. 

This follows at once from theorem VII, 155. The bundle of 
hyperplanes is given by the equation ^a i x i +\^b i x i + p^cp^O. 

III. Four hyperplanes not in the same bundle intersect in a point. 
All hyperplanes through the same point form a three-dimensional extent, 
and any four of them not in the same bundle determine the point. 



366 FOUR-DIMENSIONAL GEOMETRY 

This follows from the fact that the four equations 



determine in general a single point. The exceptions are when the 
four equations represent hyperplanes of the same bundle. 

IV. A plane and a hyper plane intersect in a straight line unless the 
plane lies entirely in the hyperplane. 

For the equations which determine the points common to a plane 
and a hyperplane are three linear equations which in general deter 
mine a line. If, however, the plane lies in the hyperplane, -the lat 
ter may be taken as one of the equations of the plane (theorem I), 
and we have only two equations. Furthermore, if the plane inter 
sect the hyperplane in three points not in the same straight line, 
it lies entirely in the hyperplane by theorem II, 155. 

V. Two planes intersect in a single point unless they lie in the same 
hyperplane. In that case they intersect in a line, or coincide. 

For the points common to two planes must in general satisfy 
four linear equations and hence reduce to a single point. If, how 
ever, the planes are in the same hyperplane, the equation of that 
hyperplane may be taken as one of the equations of each of the 
planes, and the points common to them have only to satisfy three 
equations. Furthermore, if the two planes intersect in a line, the 
hyperplane determined by four points, two on the line of inter 
section and one on each of the planes, will contain both planes 
(theorem II, 155). 

VI. Three planes not in the same hyperplane do not in general inter 
sect, but may intersect in a single point or in a straight line. Three 
planes in the same hyperplane intersect in a point or a straight line. 

The points of intersection of three planes must satisfy six equa 
tions, which is in general impossible. If the planes are in the same 
hyperplane, however, the number of equations is reduced to at least 
four by taking the equation of that hyperplane as one of the 
equations of each of the three planes. 

But consider four hyperplanes intersecting in a point. It is 
possible in a number of ways to pair these hyperplanes so as to 
determine three planes which have the point in common but are 
not in the same hyperplane. 



POINT COORDINATES 367 

Or, again, consider any two planes intersecting in a point A. 
It is easily possible to select two points B and C which shall not 
lie in the same hyperplane with either of the given planes. The 
plane ABC has the point A in common with the first two planes, 
but they do not lie in the same hyperplane. 

Similarly, let two planes intersect in a line AB. A plane may be 
passed through AB and a point C not in the same hyperplane with 
the first two planes. Of course any two of these planes lie in the 
same hyperplane (theorem V). 

VII. A straight line and a hyperplane intersect in a single point 
unless the line lies entirely in the hyperplane. 

The reason is obvious. 

VIII. A straight line and a plane do not intersect unless they lie 
in the same hyperplane. In the latter case they either intersect in a 
point or the line lies entirely in the plane. 

The points common to a straight line and a plane must satisfy 
five equations, which is in general impossible. If, however, the line 
and plane are in the same hyperplane, the number of equations 
may be reduced to four. 

Again, let the line and plane intersect in the point A. Three 
other points may be taken : B on the line, and C, D on the plane. 
The hyperplane determined by A, B, C, D then contains both the 
line and the plane. 

IX. Two straight lines do not intersect unless they lie in the same plane. 
In the latter case they intersect in a point or coincide throughout. 

The points common to two lines must satisfy six equations, 
which is in general impossible. If, however, they lie in the same 
plane, the number of equations may be reduced to four. 

Again, let the two lines intersect in a point A. The plane deter 
mined by A and two other points, one on each line, contains both 
lines. 

We close this section with two theorems on the determination 
of planes and hyperplanes which have already been foreshadowed. 

X. A plane may be determined ly (1) three points not in the same 
line ; (2) a line and a point not on it; (3) two intersecting lines. 



368 FOUR-DIMENSIONAL GEOMETRY 

XI. A hyperplane may be determined % (1) four points not in the 
same plane ; (2) a plane and a point not on it; (3) a plane and a 
line intersecting it ; (4) two planes intersecting in a line ; (5) three 
lines not in the same plane intersecting in a point. 

157. Euclidean space of four dimensions. We shall consider now 
a four-dimensional space in which metrical properties analogous to 
those of three-dimensional Euclidean space are assumed. For that 
purpose let us replace the ratios x^i x 2 : x & : x^i x 5 by x:y.z:w:t, 
and place 

JL = > Y = , Z/ = - , fr 

t t t t 

Then if t = the coordinates X, Y, Z, W are finite, and the values 
(X, Y, Z,W) are said to represent a point in finite space. If t = 
one or more of the coordinates X, Y, Z, W is infinite, and the ratios 
x : y : 2 : w : are said to represent a point at infinity. 

The equation t = Q represents, then, the hyperplane at infinity. 

The distance between two points is defined by the equation in 
the nonhomogeneous coordinates 



or in the homogeneous coordinates 



from which it appears that the distance between two finite points 
is finite and that the distance between a finite point .and an infinite 
point is in general infinite. 

The equations of a straight line are in nonhomogeneous coor 
dinates 



, V , , , , , lu 

TnoT J TT^- TT^ -TTT- 

- X-X. Y-Y, Z-Z. W-W, 

whence follows -- = - - = - - = - ^ , (4^ 

X,-X, Y 2 -Y, Z 2 -Z, Wt-wC 

which may be written 

X ~X. = Y-Y, Z-Z, W-W, 
A B C D 

The ratios A:B:C:D are independent of the two points used 
to determine the line and will be defined as the direction of the 



POINT COORDINATES 369 

line. It is readily seen that a line may be drawn through the 
point (Xj, Y^ Z^ W^) with any given direction and that two lines 
through that point with the same direction coincide throughout. 
There is, therefore, a one-to-one relation between the lines drawn 
through a fixed point and the ratios we have used to define direction. 
This justifies the use of the word. 

Two lines with the directions AjBjC^D^ and AjB^.C^D^ 
respectively are said to make with each other the angle #, defined 
by the equation 

6 



Consider the hyperplane W 0. Any point in that hyperplane 
is fixed by the coordinates (X, F, Z), and the distance between 
two points reduces to the Euclidean distance. The equation of 
any straight line in that hyperplane is 

X-X l _Y-Y l _Z-Z l 
A B C 

so that D = 0. Hence the definitions of distance and angle become 
those of Euclidean distance and angle. Therefore the geometry in 
the hyperplane W = is Euclidean. 

Similarly, the geometry in each of the hyperplanes X 0, F= 0, 
Z = is Euclidean. The same will be shown later to be true for any 
hyperplane except the hyperplane at infinity and certain exceptional 
imaginary hyperplanes. We accordingly call this four-dimensional 
geometry Euclidean. 

In the hyperplane at infinity, t = 0, a point is fixed by the 
homogeneous coordinates xiyiziw, and we may apply to this 
plane the methods and formulas of three-dimensional geometry 
with quadriplanar coordinates. 

It is important to notice the connection between figures in the 
four-dimensional space and their intercepts with the hyperplane at 
infinity. These intercepts we shall sometimes call traces. 

The equation (5) of a straight line with direction A : B : C : D may 
be written in homogeneous coordinates as 



D 



370 FOUR-DIMENSIONAL GEOMETEY 

whence it appears at once that its intercept with t is the 
point AiBiC .D. 

The equation of a hyperplane is 

Ax+By + Cz+Dw+Et = 0, 
and its trace on the hyperplane at infinity is the plane 

Ax +By 4- Cz +Dw = 0. 
Similarly, the equations of a plane are 

Ap 4- By + Cf + DJJD 4- Ef = 0, 
A 2 x 4- B z y + C 2 z 4- D 2 w + EJ = 0, 

and its trace on the hyperplane at infinity is the straight line 
Ap+Bj + Cf +&&**$, 
A 2 x 4- B$ 4- C 2 z 4- D 2 w = 0. 

A hi/persphere is defined as the locus of points whose distances 
from a fixed point are equal. It is easy to show from (2) that 
the equation of a hypersphere is 

a o (^ 2 + *, 2 + z 2 +^ 2 )4- 2 0^4- 2 a y*4- 2 a^+ 2 ajvt+af= 0, (8) 

and that its intercept with the hyperplane at infinity is the quadric 

surface 2 2 2 2 A 

ar+^-f r+r=0. (9) 

This surface, which we call the absolute, plays a role in four- 
dimensional geometry analogous to that played by the imaginary 
circle at infinity in three-dimensional geometry. All hyperspheres 
contain the absolute. The hyperplane w = intersects the absolute 
in the imaginary circle at infinity in the space of the coordinates 
#, y, z. The same thing is true of all hyperplanes, with the 
exception of the minimum hyperplanes, to be considered later. 

158. Parallelism. Any two of the configurations, straight line, 
plane, or hyperplane, are said to be parallel if their complete 
intersection is at infinity. 

This definition gives us nothing new concerning parallel lines. 
For example, we have, at once, the following theorem: 

/. Through any point in space goes a single line parallel to a fixed line. 
Any two parallel lines lie in the same plane and determine the plane. 



POINT COORDINATES 371 

Neither do we find anything new concerning a line parallel to a 
plane. We have already seen that a line will not meet a plane 
unless it lies in the same hyperplane. In the latter case the line 
may intersect the plane in a finite point or be parallel to it. We 
have the following theorem: 

//. If a line is parallel to a plane the two lie in the same hyper- 
plane and determine that hyperplane. Through any point in space 
goes a pencil of lines parallel to a fixed plane. 

When we consider parallel planes we have to distinguish two 
cases. Two planes are said to be completely parallel if they in 
tersect in a line at infinity, and are said to be simply parallel 
if they intersect in a single point at infinity and in no other 
point. 

From theorem XI, (4), 156, we have, at once, the theorem: 

III. If two planes are completely parallel they lie in the same 
hyperplane. 

In fact, completely parallel planes are the parallel planes of the 
ordinary three-dimensional geometry. On the other hand, two 
simply parallel planes do not lie in the same hyperplane and con 
sequently cannot appear in three-dimensional geometry. A dis 
tinction between completely and simply parallel planes is brought 
out in the following theorem : 

IV. If two planes are completely parallel, any line of one is parallel 
to some line of the other and, in fact, to a pencil of lines. If two 
planes are simply parallel, there is a unique direction in each plane 
such that lines with that direction in either plane are parallel to lines 
with the same direction in the other, but lines with any other direction 
in one plane are parallel to no lines of the other. 

To understand this theorem note that if two completely parallel 
planes intersect in the line I at infinity, any line in one plane will 
meet I in some point P, and any line through P in the second 
plane will be parallel to the first plane. If, however, two simply 
parallel planes intersect in a single point P at infinity, the only 
lines in the two planes which are parallel are those which intersect 
in P. It may be noticed that this property of a unique direction 



372 FOUR-DIMENSIONAL GEOMETRY 

is found also in two intersecting planes, the unique direction being 
that of the line of intersection. 

A plane is parallel to a hyperplane if they intersect in a straight 
line at infinity. Let this line be I. Then any line in the plane 
meets I in a point P, and a bundle of lines may be drawn in 
the hyperplane through P. Then each line of the bundle is par 
allel to the given line. The hyperplane meets the plane at infinity 
in a plane m, in which the line / lies. Any plane in the hyperplane 
intersects m in a line V, which has at least one point in common 
with / but which may coincide with I. From these considerations 
we state the theorem: 

V. If a plane and a hyperplane are parallel, any line in the plane 
is parallel to each line of a bundle in the hyperplane, and any plane 
in the hyperplane is at least simply parallel to the given plane. 

Two hyperplanes are parallel if they intersect in the same plane 
at infinity. Let that plane be m. Any plane in one hyperplane 
meets m in a straight line Z, and through I may be passed a pencil 
of planes in the other hyperplane. Again, consider any two planes, 
one in each of the hyperplanes. They meet m in two lines, I and I , 
which intersect in a point unless they coincide. The two planes 
can have no other point in common unless they are in the same 
hyperplane. Hence we have the theorem: 

VI. If two hyperplanes are parallel, any plane of one is completely 
parallel to some plane and hence to a pencil of planes of the other, 
and any plane of one is simply parallel to any plane whatever of 
the other to which it is not completely parallel. 

The analytic conditions for parallelism are easily given. The 
necessary and sufficient condition that two lines with the directions 

A l :B l :C^D l and A,:BjC n :D, 

should be parallel is that A^B^C^D^AjBjCjD^. 

Also the necessary and sufficient condition that two hyperplanes 

AjX+By + C\z + Dj +EJ, = 
and A 2 x +Bg + C 2 z + Djv + EJ, = 

should be parallel is that ABCDABCD 



POINT COORDINATES 



373 



Since two planes are simply parallel when they intersect in a 
single point at infinity, the necessary and sufficient condition that 



the two planes 



and 



should be simply parallel is that 






tAp + By + 6> + Dp + EJ, = 0, j 

2 =o,j 

= J 



C 



(2) 
(3) 



but that not all the other fourth-order determinants of the matrix 



A, B l 



Z> 



A ^2 ^ ^2 ^2 



should vanish. 

That the two planes (1) and (2) should be completely parallel 
their traces on the hyperplane at infinity must coincide. Now the 
determinants of the matrix 



are Pliicker coordinates for the trace of the plane. Therefore the 
necessary and sufficient condition that the two planes (1) and (2) 
should be parallel is that the determinants of the matrix 

A, B. C, 1 



should have a constant ratio to the corresponding determinants of 
the matrix 



159. Perpendicularity. In accordance with (6), 157, two lines 
with the directions A l iBjCjD l and AjBjC^D^ are said to be 
perpendicular when 

Q. (1) 



374 FOUR-DIMENSIONAL GEOMETRY 

This condition may be given a useful interpretation in the hyper- 
plane at infinity. The polar plane of a point x^y l izjw l in the 
hyperplane t = 0, with respect to the absolute x 2 + y 1 + 2 2 + w 2 = 0, is 
xjc + y^y -{-z^ + w^w = Q. Equation (1) therefore shows that two 
perpendicular lines meet the hyperplane at infinity in two points, 
each of which is on the polar plane of the other with respect to 
the absolute. Or, otherwise expressed, the necessary and sufficient 
condition that two lines are perpendicular is that their traces on the 
hyperplane at infinity are harmonic conjugates with respect to the two 
points in which the line connecting the traces meets the absolute. 

A line is said to be perpendicular to a hyperplane when it is 
perpendicular to every line in the hyperplane. For this to happen 
it is necessary and sufficient that the hyperplane meet the hyper 
plane at infinity in the polar plane of the trace of the line. From 
this follows at once the theorem: 

/. Through any point either in or without a hyperplane one and 
only one straight line can be drawn perpendicular to the hyperplane ; 
and from any point in or without a straight line one and only one 
hyperplane can be drawn perpendicular to it. 

Since in the plane at infinity the polar plane with respect to the 
absolute of the point A:B:C:D is the plane Ax +By + Cz + Dw = 0, 
we have the theorem: 

//. Any line perpendicular to the hyperplane Ax+By-\- Cz-\-Dw+E= 
has the direction A:B:C:D, and conversely. 

Any three lines of a hyperplane which are not coplanar, and no 
two of which are parallel, determine three noncollinear points of 
the trace of the hyperplane at infinity. The line perpendicular to 
these three lines passes through the pole of the plane determined 
by the three points. Consequently we have the theorem : 

///. A line perpendicular to three lines of a hyperplane which are 
not coplanar, and no two of which are parallel, is perpendicular to 
the hyperplane. 

In particular the three lines may intersect in the same point. 
Consequently we have the theorem: 

IV. A line may be drawn perpendicular to three lines intersecting 
in a point but not in the same plane, and it is then perpendicular to 
the hyperplane determined by the three lines. 



POINT COORDINATES 375 

A line is perpendicular to a plane if it is perpendicular to every 
line in that plane. From this we have the theorem: 

V. If a line is perpendicular to a hyperplane, it is perpendicular 
to every plane in the hyperplane. 

The definition of perpendicularity of line and plane is the same 
as in three-dimensional geometry. The theorem, however, that from 
a point in a plane only one line can be drawn perpendicular to it is 
no longer true. 

In fact, consider a plane / and any point P in it, and let the 
trace of I on t = be the line L. Further, let L be the conjugate 
polar line of L with respect to the absolute ( 92). Then any point 
on L is the harmonic conjugate of any point on L. Hence any two 
lines, one of which intersects L and the other L 1 , are perpendicular. 
From P a pencil of lines may be drawn to meet L . Therefore we 
have the theorem: 

VI. All lines perpendicular to a plane at a fixed point lie in a plane. 
The two planes are such that every line of one is perpendicular to 
every line of the other. 

These planes are said to be completely perpendicular. Obviously 
they do not exist in ordinary three-dimensional space. 

The point P considered above need not lie in the plane I. Hence 
we have the more general theorem: 

VII. Through any point of space one plane, and only one, can be 
passed completely perpendicular to a given plane. 

With the same notation as before let I be a given plane, P a 
point which may or may not lie in I, and PA a line perpendicular 
to I, where A lies on L . Through PA pass a plane m intersecting 
t = in a line M through A. If M is the conjugate polar of M, 
M f intersects L in a point B, by the theory of conjugate polar lines. 
Then if Q is any point of I, the line QB lies in I and is perpen 
dicular to m. Therefore we have the following theorem : 

VIII. If a plane m contains a line perpendicular to a plane I, the 
plane I contains a line perpendicular to m. 

Two planes such that each contains a line perpendicular to the 
other we shall call semiperpendicular planes. 



376 FOUR-DIMENSIONAL GEOMETRY 

From the foregoing we easily deduce the following theorem : 

IX. The necessary and sufficient condition that two planes should be 
semiperpendicular is that the trace at infinity of either should intersect 
in one point the conjugate polar with respect to the absolute of the trace 
of the other. The necessary and sufficient condition that two planes 
should be completely perpendicular is that the trace of either should 
be the conjugate polar of the trace of the other. 

If two semiperpendicular planes lie in the same hyperplane, 
they intersect in a line and are the ordinary perpendicular planes 
of three-dimensional geometry. If two semiperpendicular planes 
are not in the same hyperplane, they intersect in a single point. If 
this point is at infinity, the two planes are also simply parallel. 
In these cases the traces L and M intersect in a point C, which is 
harmonic conjugate to both A and B. From this follows the 
theorem : 

X. Two semiperpendicular planes may be simply parallel. The 
direction of the parallel lines of the two planes is then orthogonal to 
the directions of the perpendicular lines. 

It is to be noticed that in this case the direction of the parallel 
lines is similar to that of the line of intersection of semiperpen 
dicular planes in the same hyperplane. 

A plane I is perpendicular to a hyperplane h when it contains 
a normal line to the hyperplane. The trace L of the plane then 
passes through the pole of the trace H of the hyperplane, and the 
conjugate polar L of L lies in H. Therefore : 

XI. If a plane is perpendicular to a hyperplane, it is completely 
perpendicular to each plane of a pencil of parallel planes of the hyper 
plane and semiperpendicular to every other plane of the hyperplane. 

The angle between two hyperplanes may be denned as the angle 
between their normal lines. Hence two hyperplanes, 

Ajc+By + Cf+Djo+EJ = 
and A z x + By + Of + D^w + EJ = 0, 

are perpendicular when and only when 



POINT COORDINATES 377 

This is the condition that the traces at infinity of the two hyper- 
planes are such that each contains the pole of the other, as might 
be inferred from the definition. From this we have the theorems : 

XII. If two hyperplanes are perpendicular, the normal to either from 
any point of their intersection lies in the other. 

XIII. Any hyperplane passed through a normal to another hyper- 
plane is perpendicular to that hyperplane. 

Since in t = the intersection of two planes is the conjugate 
polar of the line connecting the poles of the planes, we have the 
theorem : 

XIV. The plane of intersection of two perpendicular hyperplanes is 
completely perpendicular to any plane determined by two intersecting 
normals to the hyperplanes. 

In the hyperplane at infinity we may, in an infinite number of 
ways, select a tetrahedron ABCD which shall be self -con jugate with 
respect to the absolute. From any finite point draw the lines 
OA, OB, OC, OD. We have a configuration, the properties of which 
are given in the following theorem : 

XV. From any point in space may be drawn, in an infinite number 
of ways, four mutually perpendicular lines. Every three of these lines 
determines a hyperplane perpendicular to the hyperplane determined 
by any other three. Every pair of the lines determines a plane which 
is completely perpendicular to that determined by the other pair of 
the lines. 

A special case of the configuration described above is that formed 
by the coordinate hyperplanes X= 0, Y= 0, Z= 0, W= 0. 

By (6), 157, the cosines of the angles made with the coordi 
nate hyperplanes by the hyperplane 

Ax + By 4- Cz + Div +E= 
A B 



are 



C 



when 



378 FOUR-DIMENSIONAL GEOMETRY 

We may denote these by Z, m, n, r respectively, and write the 
equation of the hyperplane in the form 

Ix 4- my 4- nz + rw -f p = 0, 

with 1 2 + w 2 -f- n 2 + r 2 = l. The equation is then in the normal form, 
and it is easy to show that p is the length of the perpendicular 
from the origin to the plane. Also by the same methods as in 
three-dimensional geometry we may show that the length of the 
perpendicular from any point (x^ y^ z^ w^) is Ix^+my^ nz^-\- rw^+p. 
Let us now take any configuration described in theorem XV, 
and, writing the equation of each of the four hyperplanes in the 
normal form, make the transformation of coordinates given by the 
equations in nonhomogeneous coordinates: 

x = l^x + m^y + n^ + r^w + p^ 
y = Ip + my + n 2 z + r 2 w +p z , 



4- n 
with the conditions ZJ+ wij-J- nf -f rj =1, 

l} k + m i m k + w t % H- r i r k = 0. 



The new coordinates are the distances from four orthogonal hyper- 
planes, and, in fact, our discussion shows that the same is true of 
the original coordinates. 

In the new system the equation for distance is unaltered, namely, 



and if we place w =Q we have the ordinary Euclidean geometry 
in three dimensions. This justifies the statement already made in 
anticipation, which we now give as a theorem : 

XVI. In four-dimensional Euclidean space the geometry in any 
hyperplane, for which A* + B 2 + C 2 + D 2 = 0, is that of the usual 
three-dimensional Euclidean geometry. 

160. Minimum lines, planes, and hyperplanes. In the discussion 
of the previous section we have had to make exception of the cases 
in which the direction quantities A, B, C, D satisfy the condition 

:0. (1) 



POINT COORDINATES 379 

We shall now examine the exceptional eases. 

Obviously the necessary and sufficient condition that the direction 
quantities of a straight line satisfy equation (1) is that the line inter 
sects the absolute, or, in other words, that the trace at infinity of the 
line lies on the absolute. The necessary and sufficient condition that 
the quantities A, B, C, D in an equation of a hyperplane satisfy (1) is 
that the trace at infinity of the hyperplane is tangent to the absolute. 
In this case the hyperplane is said to be tangent to the absolute. 

The straight lines which intersect the absolute are the minimum 
lines of three-dimensional geometry. 

In fact, the hyperplane w = 0, which by theorem XVI, 159, 
represents any ordinary hyperplane, meets the absolute in the imag 
inary circle at infinity, and the lines in the hyperplane which meet 
the absolute are therefore the minimum lines of the hyperplane. 
Also, if any line meets the absolute in a point P, a hyperplane 
can evidently be determined in an infinite number of ways so as 
to contain the line and not be tangent to the absolute. We have, 
therefore, nothing new to add to the three-dimensional properties 
of minimum lines. 

In four-dimensional space there go through every point co 2 mini 
mum lines, one to each of the points of the absolute. These lines 
form a hypercone. A hyperplane through the vertex intersects the 
hypercone in general in an ordinary cone of minimum lines, and a 
plane through the vertex intersects the hypercone in general in two 
minimum lines. 

Consider now any plane. Its trace in the hyperplane at infinity 
is a straight line which may have any one of three relations to 
the absolute : (1) it may intersect the absolute in two distinct 
points ; (2) it may be tangent to the absolute ; (3) it may lie 
entirely on the absolute. 

The first case is the ordinary plane, the second the minimum 
plane of three-dimensional geometry. In fact, if any plane of 
character (1) or (2) is given, it is clearly possible to find a hyper 
plane which will contain it and not be tangent to the absolute. 
The ordinary plane is characterized by the property that through 
any point of it go two minimum lines, and the minimum plane of 
three-dimensional type by the property that through every point 
of it goes one minimum line. 



380 FOUR-DIMENSIONAL GEOMETRY 

The third type of plane is, however, not found in the ordinary 
three-dimensional geometry. For if a plane meets the absolute in 
a straight line, any hyperplane containing it contains this line and 
therefore intersects the absolute in two straight lines. The geometry 
in this hyperplane is therefore a geometry in which the imaginary 
circle at infinity is replaced by two intersecting straight lines. Its 
properties will therefore differ from those of Euclidean space. 

A plane at infinity intersecting the absolute in two straight lines 
is tangent to it. Therefore a plane of the third type lies only in 
hyperplanes tangent to the absolute. A unique property of these 
planes is that any straight line in them meets the absolute and is 
therefore a minimum line. In other words, the distance between 
any two points on planes of this type is zero. We shall refer to a 
plane of this type as a minimum plane of the second kind. 

Consider now a hyperplane which is tangent to the absolute. 
The equation of such a hyperplane is 

Ax +By + Cz +Dw +E= 

with ^4 2 -f-I? 2 + (7 2 -f-D 2 = 0. From analogy to three-dimensional 
geometry we shall call such a hyperplane a minimum hyperplane. 
It has already been remarked that in a minimum hyperplane we 
have at infinity two intersecting straight lines instead of an imagi 
nary circle. There will be a unique direction in the hyperplane; 
namely, that toward the point of intersection of the two imagi 
nary lines at infinity. For convenience we shall call a line with 
this direction an axis of the hyperplane. 

Through every point of the hyperplane goes an axis, and through 
every axis go two minimum planes of the second kind, each con 
taining one of the two intersecting lines at infinity. Any other 
plane through the axis is an ordinary minimum plane. The cone 
of minimum lines through a point splits up, then, into two inter 
secting planes. 

Any plane not containing the axis intersects the absolute in two 
distinct points and is therefore an ordinary plane. 

Since a minimum hyperplane intersects t = in a plane tangent 
to the absolute, the normal to the hyperplane passes through the 
point of tangency, which is the point of intersection of the two 
straight lines at infinity. Hence the axes of a minimum hyperplane 




POINT COORDINATES 381 

are the normals to the hyperplane. The axes are therefore normal 
also to every plane in the minimum hyperplane. 

Let the plane of the figure (Fig. 60) be the plane of intersection 
of a minimum hyperplane with the hyperplane at infinity, and let 
the two lines OA and OB be the intersection of the plane with the 
absolute. Then, if L is the trace of any ordinary plane, the normal 
to the plane passes through and is an axis of the hyperplane. 
Two ordinary planes in the minimum 
hyperplane, therefore, cannot be per 
pendicular to each other. 

But consider a minimum plane of 
the first kind whose trace on the hyper 
plane at infinity is the line OQ. The 
conjugate polar of the line OQ is a line 
OR. Consequently any two minimum 
planes of the first kind whose traces 
are OQ and OR respectively are com 
pletely perpendicular. This state of FIG 6Q 
two completely perpendicular planes 

lying in the same hyperplane cannot be met in an ordinary hyper 
plane and is therefore not found in Euclidean geometry. This 
is due to the fact that in an ordinary hyperplane only one mini 
mum plane can be passed through a minimum line, while in a 
minimum hyperplane a pencil of minimum planes can be passed 
through an axis of the hyperplane, and these planes are paired 
into completely perpendicular planes. 

Finally, it may be remarked that a minimum plane of the second 
kind is, in a sense, completely perpendicular to itself, for the lines 
OA and OB are each self -conjugate. 

For the sake of an analytic treatment let us suppose that a 
minimum hyperplane has the equation z i w = 0, and let us make 
the nonorthogonal change of coordinates expressed by the equations 

z = z + iw, 
w f = z iw. 

Then the formula for distance becomes 



382 FOUR-DIMENSIONAL GEOMETRY 

In the hyperplane w = a point is fixed by the coordinates 
x, y, z\ and the distance between two points becomes 



The equation of the two straight lines at infinity is 



and the equations of any axis of the hyperplane is x = x^ y = y Q . 

In the formula for distance the coordinate z does not occur. 
Hence the distance between two points is unaltered by displacing 
either of them along an axis. 

Consider the equation 



This represents the locus of points at a constant distance a from 
a fixed point x Qj y^ z, where z is arbitrary. From the form of the 
equation the locus is a cylinder whose elements are axes. Every 
point on the cylinder is at a constant distance a from each point 
of the axis x = # , y y^. 

The above are some of the peculiar properties of a minimum 
hyperplane. 

161. Hypersurfaces of second order. Consider the equation 

2)0*% = OH = a ik) CD 

in the homogeneous coordinates of a four-dimensional space in 
which no hyperplane is singled out to be given special significance 
as the hyperplane at infinity. The space is, therefore, a projective 
space. The student will have no difficulty in showing, by the methods 
of 82, that the coordinates may, if desired, be interpreted as 
equal to the distances from five hyperplanes, each distance multi 
plied by an arbitrary constant. However, we shall make no use of 
this property, and mention it only for the analogy between the present 
coordinates and quadriplanar coordinates in three-dimensional space. 
Equation (1) represents a hypersurface of the second order. If 
y { and z i are any fixed points, the line 

px^y.+ tet (2) 

intersects the hypersurface in general in two distinct or coincident 
points or lies entirely on it. Therefore any hyperplane intersects 
the hypersurface in a two-dimensional extent which is met by any 



POINT COORDINATES 383 

line in two points and is therefore a quadric surface, or else the 
hyperplane lies entirely on the hypersurface. Similarly, any plane 
intersects the hypersurface (1) in a conic or lies entirely on it. 

Let us consider these intersections more carefully. If in equa 
tion (2) the point y. is taken on the hypersurface, the line will meet 
the hypersurface (1) in two distinct points unless the equation 

2<W*= (3) 

is satisfied by the point z t . In the latter case the line (2) meets 
(1) in two points coinciding with y { , unless also z. is on the hyper 
surface, in which case the line lies entirely on the hypersurface. 

This means that if y ( is on the hypersurface (1), any point on 
the hyperplane 2<W t =0 (4) 

but not on the hypersurface, if connected with y^ determines a 
straight line tangent to the hypersurface, and this property is 
enjoyed by no other point. Hence the hyperplane is the locus of 
tangent lines at y i and is called the tangent hyperplane. 

The hyperplane (4) intersects the hypersurface in an extent of 
two dimensions which has the property that any point on it deter 
mines with y. a line entirely on it. It is therefore a cone of second 
order. Therefore, through any point of the hypersurface goes a cone 
of straight lines lying entirely on the hypersurface. 

An exception to the above occurs when y { is a point satisfying 
the equations ^^ a ^ + a ^ + a ^ + a ^ = . (5) 

Such a point, if it exists, is a singular point. At a singular point 
the equation of the tangent hyperplane becomes illusive. Any line 
through a singular point meets the hypersurface in two coincident 
points, and if any point on the hypersurface is connected with the 
singular point by a straight line, the line lies entirely on the hyper 
surface. Equations (5) do not always have a solution ; but if they 
have, the solution is a point of the surface, since equation (1) is 
homogeneous. 

If y i is any point, whether on the hypersurface or not, equation (4) 
defines a hyperplane called the polar hyperplane of y.. If the equation 
of the polar hyperplane is written in the form 

u i x i + V* + Va + V 4 + u s x s = 
we have pu t = a^, + a ia y 3 + a i9 y 9 + a i4 y 4 + a i6 y & . (6) 



384 FOUR-DIMENSIONAL GEOMETRY 

From this it follows that any point has a definite polar hyper- 
plane. The converse is true, however, only if the determinant 



does not vanish. The vanishing of this determinant is the necessary 
and sufficient condition that equations (5) should have a solution. 
Therefore we say: 

If a hyperplane of the second order has no singular points, to every 
point in space corresponds a unique polar hyperplane, and to every 
hyperplane corresponds a unique pole. The necessary and sufficient con 
dition for this to occur is that the discriminant \a ik \ should not vanish. 

If the hypersurface has a singular point, it is easy to see that 
every polar hyperplane passes through that point. Therefore only 
hyperplanes through the singular points can have poles. 

The properties of polar hyperplanes are similar to those of polar 
planes of three-dimensional geometry, arid the theorems of 92 may, 
with slight modifications, be repeated for the four dimensions. 

We may also employ some of the methods of 93 in classi 
fying hypersurfaces of the second order. Let us take the general 
case in which no singular points occur. There is then no difficulty 
in applying these methods to show that the equation may be 
reduced to 



The cases of hypersurfaces with singular points are more tedious 
if the elementary methods are used. It is preferable in these cases 
to use the methods of elementary divisors. 

162. Duality between line geometry in three dimensions and point 
geometry in four dimensions. Since the straight line in a three- 
dimensional space is determined by four coordinates, it will be 
dualistic with the point in four dimensions. In order to have 
coordinates of the four-dimensional space which are dualistic with 
the Klein coordinates of the straight line, we will introduce hexa- 
spherical coordinates in four-dimensional space analogous to the 
pentaspherical coordinates of three-dimensional space. 



POINT COORDINATES 385 

Following the analogy of 117, 123, let us place 



= 22T, 



(2) 
where x* + x* + xl -h xf -+- x* + x* = 0. 

The coordinates x t are hexaspherical coordinates. The locus at 
infinity has the equation x l -f ix^ = 0, and the real point at infinity 
has the coordinates 1 : : : : : i. 

The equation 



is that of the hypersphere 



There are four varieties of hyperspheres : 

1. Proper hyperspheres, ^af^O, 

2. Proper hyperplanes, ^ 2 =^0, 

3. Point hyperspheres, V 2 =0, 

4. Minimum hyperplanes, 2 2 = 0, 



On the other hand, we may interpret the coordinates x i as Klein 
coordinates of a straight line in a space of three dimensions. 

For convenience we will denote by S B the three-dimensional 
point space in which x { are line coordinates, and by 2 4 the four- 
dimensional point space in which x i are hexaspherical coordinates 
of a point. Then the coordinates 1:0:0:0:0:^, which in 2 4 
represent the real point at infinity, represent in S 8 a straight line Z, 
which has no peculiar relation to the line space. In fact, I acquires 
its unique significance only because of its dualistic relation to S 4 . 
Also the equation 2^+12^=0, which, in 2 4 , represents the hyper- 
plane at infinity, represents in S 3 a special line complex c, of which 
the line I is the axis. With these preliminary remarks we may 
exhibit in parallel columns the relation between S 8 and 2 i . 



386 



FOUR-DIMENSIONAL GEOMETRY 



Point. 

Real point at infinity. 

Proper hypersphere. 

Proper hyperplane. 

Point hypersphere. 

Center of point hypersphere. 

Minimum hyperplane. 

Hyperplane at infinity. 

Two points on same minimum 
line. 

Any imaginary point at infinity. 

Points common to two hyper- 
spheres. 

Vertices of two point hyper- 
spheres which pass through the 
intersections of two hyperspheres. 

Circle defined by the intersection 
of three hyperspheres. 

Two circles such that each point 
of one is the center of a point hyper 
sphere passing through the other. 



T 3 

Line. 
Line I. 

Nonspecial line complex not con 
taining I. 

Nonspecial complex containing I. 
Special complex not containing I. 
Axis of special complex. 
Special complex containing I. 
Special complex c with axis I. 
Two intersecting lines. 

Line intersecting I. 
Line congruence. 

Axes of line congruence. 



Regulus. 

Two reguli generating the same 
quadric surface. 



The use of hexaspherical coordinates gives a four-dimensional 
space in which the ideal elements differ from those introduced by 
the use of Cartesian coordinates, as has been explained in 123. 
Such a space is in a one-to-one relation with the manifold of straight 
lines in g . 

If we wish to retain in 2 4 the ideal elements of the Cartesian 
geometry, the relation between S 8 and 2 4 ceases to be one-to-one for 
certain exceptional elements. To show this we will modify equa 
tions (1) by introducing homogeneous coordinates in 2 4 and have 



= 2 xt , 



POINT COORDINATES 387 

If we use these equations to establish the relation between the 
lines of S s and the points of 2 4 , we shall have the same results 
as before, with the following exceptions, all of which relate to the 
ideal elements of 2 4 . Any point in 2 4 on the hyperplane at infinity, 
but not on the absolute, corresponds to the line I ; and the line I 
corresponds to all points on t = 0, but not on the absolute. 

Any point on the absolute corresponds to a line in $ 3 which at 
first sight seems entirely indeterminate, but if we write equations 
(3) in the form 



it appears that a point on the absolute corresponds to a line for 
wnicn o^ : # 6 = 1 : z, x^\x^.x^\x b = x\y\z\w. 

This is a one-dimensional extent of lines. One line of the extent 
is always /, and another is \\x\y\z\w.i. The general line may 
be written as (1 4- X): x : y : z : w : i(l + X). By 131 the extent is, 
therefore, a pencil containing I. Then, to any point on the absolute 
corresponds any line of a certain pencil containing /. 

It is easy to show that any line which intersects I corresponds 
to a definite point on the absolute. 

It is, of course, possible to interpret equation t = in equations (3) 
as the equation of any hyperplane in a protective space with the 
coordinates x : y : z : w : t. The absolute is then replaced by a quadric 
surface <E> in the hyperplane t = 0. The correspondence between 
S s and 2 4 is then less special than the one we have considered. 

EXERCISES 

1. Show that orthogonal hyperspheres correspond to complexes in 
involution. 

2. Define inversion with respect to a hypersphere F in 2 4 and show 
that two inverse points with respect to F correspond to two lines in 
5 8 which are conjugate polars with respect to the line complex which 
corresponds to F. 



CHAPTER XX 

GEOMETRY OF N DIMENSIONS 



163. Projective space. We shall say that a point in n dimensions 
is defined by the n ratios of n + I coordinates; namely, 



The values of the coordinates may be real or imaginary, but the 
indeterminate ratios : : : : shall not be allowed. The 
totality of points thus obtained is a space of n dimensions de 
noted by S n . 

A straight line in S n is defined by the equations 

/>*,.= &+X2,., (i=l, 2, ...,w+l) (2) 

where y { and z t are constants and X is an independent variable. 
Obviously y. and z i are coordinates of two points on the line, which 
is thus uniquely determined by any two points in S n . Also, any 
two points of a straight line may be used to define it. 
A plane in S n is defined by the equations 

px i= y L + XZi+pWp (i =1, 2, . . ., n + 1) (3) 

where y^ z { , w i are the coordinates of three points not on the same 
straight line, and X, /A are independent variables. Therefore a plane 
is uniquely determined by any three noncollinear points of S n , and 
any three such points on a plane may be used to define it. 

In general, a manifold of r dimensions lying in S n may be defined 
by the equations 



, (1=1,2,..., *+l) (4) 

where y. are constants not connected by linear relations of the same 
form as (4), and \ k are r independent variables. Such a manifold 
is called a linear space of r dimensions and will be denoted by /. 
It is also called an r-flat. A straight line is therefore a linear space 
of one dimension (/), a plane is a linear space of two dimensions 

388 



POINT COORDINATES 389 

2), and S n itself is a linear space of n dimensions. From the 
definition follow at once the theorems : 

/. A linear space of r dimensions is uniquely determined by any 
r -h 1 points of S n not lying in a linear space of lower dimensions, and 
any r H-l points of an S r may be used to define it. 

II. A linear space of r dimensions is determined by a linear space 
of r 1 dimensions and any point not in that latter space. 

It is easy to see that a linear space of n 1 dimensions is also 
defined by a linear equation 

a^-h 2 :r 2 + . . . + a n x H + a n + ,x n + l = 0, (5) 

which is analogous to the equation of a plane in three dimensions. 
An $ _! is therefore called a hyperplane. 

It is also easy to see that the coordinates x f which satisfy equa 
tions (4) satisfy n r equations of the form (5), and conversely. 
Therefore 

///. A linear space of r dimensions may be defined by n r inde 
pendent linear equations, and is therefore the intersection of n r 
hyperplanes. 

In S n we shall be interested in projective geometry ; that is, in 
properties of the space which are unaltered by the transformation 



where the determinant \a ik \ does not vanish. Accordingly, if we 
are concerned with geometry in an S r we may equate to X r 
X r+3 , ., -3T n+1 , respectively, the left-hand members of the n 
equations which define it, while leaving x^ # 2 , , x r+l unchanged. 
Now placing X r + 2 , X r+3 , , X n + 1 equal to zero, we have left the 
r -hi homogeneous coordinates x v ar 2 , , x r + l to define a point in S r . 
It follows that an S, is an S n with a smaller number of dimensions, 
and that any projective properties of S n which are independent of 
the value of n apply to any S .. 

Besides the linear spaces there may exist in S n other spaces. 
Such spaces may be defined by equations of the form 



+ 2 

T 



390 ^-DIMENSIONAL GEOMETRY 

where </> t are functions of r independent variables X r If <^ are 
algebraic functions, equations (7) define an algebraic space. If we 
substitute the values of x i from (7) in the r equations, 



which define an $,[_,., we shall have r equations to determine the r 
variables \. The solutions of these equations used in (7) give the 
number of points of the space (7) which lie in an S^_ r . Let this 
number be g. Then g is called the degree of the space (7), and 
that space is denoted by $,?, where r gives the dimensions of the 
space and g the number of points in which it is cut by a general S n _ r . 
Thus Sf represents a curve which is cut by any hyperplane in g 
points, and S^_ 1 a hypersurface which is cut by any straight line 
in g points. 

A space S? may also be defined by n r simultaneous equations. 
Usually the same space may be represented by either this method 
or by that of equations (7), but sometimes this is not possible. 
If *_! is represented by a single algebraic equation, g represents 
the degree of the equation. If Sf is represented by n r equations, 
g is in general the product of the degrees of the equations. 

In this chapter we shall confine our attention to S^ defined by 
the equation 

f=n + l k=n+l 
i=l *=1 

and sections of the same. 

164. Intersection of linear spaces. Consider two linear spaces 
SJ. and S^ . A point x^ which is common to the two, must satisfy 
the 2 n r^ r 2 equations in n + 1 homogeneous variables : 

/fUXr _i_ //IXy _l_ _1_ /Y<!) /y 

"l ^i i t*o X 2 T T w n + i x -i-i v, 



POINT COORDINATES 391 

We have three cases to distinguish : 

1. If 2 n r l r 2 > n, equations (1) have in general no solution. 
There results the theorem : 

/. Two linear spaces Si and Si have in general no point in common 
when r^+r z < n. 

For an example consider two straight lines in S 3 or a straight 
line and a plane in S 4 . 

2. If 2 n r^ r 2 = n, equations (1) have in general one solution. 
There results the theorem : 

//. Two linear spaces Si and Si intersect in general in one point 
when r + 



Examples are two straight lines in $ 2 , a line and a plane in g , 
and two planes in $ 4 . 

3. If 2 n r^ r 2 < n, equations (1) have in general an infinite 
number of solutions. Let us suppose that r l -f- r 2 = n -}- a. The 
number of equations (1) is then n a, and they therefore define 
an S^. There results the theorem: 

///. Two linear spaces Si and S , where r 1 H- r r = n -f a, intersect 
in general in an S^. 

Examples of this theorem are that in S 3 two planes intersect in 
a straight line, and that in $ 4 two hyperplanes intersect in a plane. 

Of course any two linear spaces may so lie as to intersect in 
more points than the above general theorems call for. Let us sup 
pose then that Si and Si intersect in an S^. Now Si is defined by 
r i + l points, of which a+1 may be taken in S^. Similarly, Si is 
defined by r 2 + l points, of which a+1 may be taken in S^. If, 
therefore, we take a -f 1 points in S^ r^ a other points in S 1 but 
not in Sh and r 2 a points in Si but not in S^ we have r l +r 2 a+~L 
points, which may be used to define an Sl +r _ a > This Sl +r _ a con 
tains all of Si and all of Si since it contains rj + 1 points of the 
former and r 2 +l points of the latter. 

Therefore we have the theorem : 

IV. If Si and Si intersect in an S^ they lie in an Si + r _ a . 

An example of this theorem is that in S s if two straight lines 
(/) intersect in a point ( ), they lie in a plane ([). Another 



392 ^-DIMENSIONAL GEOMETRY 

example is that in 4 if two planes (, ) intersect in a straight 
line ($/), they lie in an /S 8 . 

Conversely, we have the theorem : 

V. If S and S^ lie in an S m (m < n), they intersect in an S r ^ r ^_ m 
if r^+r^m. 

This is only a restatement of theorem III, since by the previous 
section we have only to consider the S m in which the two linear 
spaces lie. 

Similar theorems may be proved for the intersections of the 
curved spaces S l and S 9 r *. These we leave for the student. 

EXERCISES 

1. Show that the hyperplanes in S n may be considered as points in a 
space of n dimensions S n . 

2. Show that if S^ contains p -f- 1 points of S^ it contains all 
points of Sp. 

3. Show that through any Si may be passed ao""*" 1 hyperplanes, 
any n k of which determine Si ; that is, in the notation of Ex. 1 any 
Si is common to a X-t-i- 

4. Show that two algebraic spaces S? n and S* do not in general 
intersect if in + m < n, and intersect in an S g a 9 if m -\- m = n -\- a. 

5. Show that every S ^ is contained in an ^ l + 1 . 

6. Show that every curve of order g is contained in a linear space of 
a number of dimensions not superior to g. 

165. The quadratic hypersurface. The equation 

i=n + l k=n+l 

C 1 ) 

defines an >SJ_ 1? which we shall call a quadratic hypersurface or, 
more concisely, a quadric. For convenience we shall denote the 
surface by </>. 

Any line px i =y i -\-\z i (2) 

meets c/> in two points corresponding to values of X given by the 
equati 2 Xa^ + X 2 a z ., % = 0. (3) 



POINT COORDINATED 393 

If ^a ik y^ t = 0, the points # t . and z i are harmonic conjugates with 
respect to the points in which the line (2) intersects <, and are called 
conjugate points. Therefore, if y { is fixed, any point on the locus 

W*=0 (4) 

is a harmonic conjugate of y.. This locus is a hyperplane called 
the polar hyperplane of y i with respect to the quadric. 

If y { is also on the quadric, both roots of (3) are zero, and the 
line (2) touches the hypersurface in two coincident points at y i9 
or lies entirely on </>. The polar (4) then becomes the tangent 
hyperplane, the locus of all lines tangent to < at y t . In no other 
case does the polar contain the point y { . 

It follows directly, either from the harmonic property or from 
equation (4), that if a point P is on the polar of a point Q, then 
Q is on the polar of P. 

More generally, let y i describe an S r defined by 

/>y,=yS"+Xtf>+...+\tf-+ >. (5) 

The polar hyperplanes are 



Values of x i common to these hyperplanes satisfy the r +1 equations 

(6) 



and therefore form an /Sj_ r _ r The two spaces S r and. S^ l _ r _ l are 
conjugate polar spaces. Each point of one is conjugate to each 
point of the other. Conjugate polar lines in $ 3 form a simple 
example. 

If the equation of the polar hyperplane is written in the form 



we have /w* = 2 a ,*y.- CO 



Let us consider first the case in which the determinant |a a |, 
which is the discriminant of (1), does not vanish. Then if the 
quantities u k in (7) are replaced by zero, the equations have no 
solution. Therefore all possible values of y. give definite values of 
u k which cannot all become zero. Again, equations (7), as they 



394 ^-DIMENSIONAL GEOMETRY 

stand, can be solved for y it so that any assumed values of u k deter 
mine unique values of y i which cannot all be zero. Summing up, 
we have the theorem: 

If the discriminant of $ does not vanish, every point of S n has a 
definite polar hyperplane, and every hyperplane in S n is the polar 
of a definite point. In particular, at every point of <f> there is a 
definite tangent plane. 

Consider now the case in which the discriminant |a ijfc | vanishes. 
There will then be solutions of the equations 

Any point whose coordinates satisfy (8) lies on </>, since its 
coordinates satisfy the equation 



and is called a singular point of $. 

Obviously, at a singular point the tangent hyperplane is indeter 
minate, and in a sense any hyperplane through a singular point 
may be called a tangent hyperplane. 

Equation (3) shows that any line through a singular point cuts 
the quadric in two points coincident with the singular point, which 
is thus a double point of the quadric. It also appears from (3) 
that any point of <f> may be joined to any singular point by a straight 
line lying entirely on <f>. 

Any point y. not a singular point has a definite polar hyperplane 

Jfc=n+l fi= n +l 



*=i 
and since this may be written 



it passes through all the singular points. 

The number of the singular points of <f> will depend upon the 
vanishing, or not, of the minors of | a ik . In the simplest case, in which 
\a ik \ vanishes but not all of its first minors vanish, equations (8) 



POINT COORDINATES 395 

have one and only one solution, and < has one singular point. 
Therefore the quadric consists of oo n ~ 2 lines passing through the 
singular point. 

Suppose, more generally, the minors of | a ik which contain n + 2 r 
or more rows vanish, but that at least one minor with n + 1 r rows 
does not vanish. The equations (8) then contain n r-fl inde 
pendent equations, and the singular points therefore form an S r _ r 
The quadric is then said to be r-fold specialized. The number r is 
so chosen that a onefold specialized quadric has a single singular 
point, a twofold specialized quadric has a line of singular points, 
and so on. 

Any S r which is determined by the S r _^ of singular points and 
another point P on <> lies entirely on <. This follows from the fact 
that all points of the S r lie on some line through P and a singular 
point, and, as we have seen, these lines lie entirely on <f>. In par 
ticular, if r = 2, the quadric consists of planes through a singular 
line ; if r = 3, the quadric consists of spaces of three dimensions 
through a singular plane ; and so forth. 

A group of w-f-1 points which are two by two conjugate with 
respect to <f> form a self -con jugate (w-J-l)-gon. There always exist 
such (n + l)-gons if the quadric is nonspecialized. This may be 
seen by extending the procedure used in 92. By a change of 
coordinates the n + 1 hyperplanes which are determined by each 
set of 7i-points in the (w-f-l)-gon may be used in place of the 
original hyperplanes x i = 0. In the new coordinates any point 
whose coordinates are of the form x k = l, # t .= (i=fc) has the 
hyperplane x t =Q for its polar. The equation of </> then becomes 

<**+< + +.*.%! = <> (9) 

Now the vanishing of the discriminant and its minors denotes 
geometric properties which are independent of the coordinates used. 
Hence we infer that for the general quadric all the coefficients c i 
differ from zero. If the quadric is r-fold specialized, it may be 
shown that equation (9) may still be obtained, but that r of 
the coefficients vanish. 

If the quadric is general, by another change of coordinates 
equation (9) may be put in the form 



396 ^-DIMENSIONAL GEOMETRY 

EXERCISES 

1. Prove that all points of any S^ through the S r _ l of singular points 
have the same polar hyperplane, which passes through Sj-i> and that, 
conversely, any hyperplane through the singular S r _ 1 has for its pole 
any point of a certain S r . 

2. Show that for any quadric which is r-fold specialized, any tangent 
hyperplane at an ordinary point is tangent to the quadric at all points 
of an $1. lying on <f> and determined by the point of contact and the 
singular S r _ l . 

3. Show that if <f> is more than once specialized, any hyperplane is a 
tangent hyperplane at one or more of the points of the singular S r _ l . 

4. Prove that every S^ through a point y { intersects < in an 8%^ and 
intersects the polar hyperplane of y { in an S n _ l} which is the polar hyper 
plane of y { with respect to the S^_ l in the space S m . 

6. Prove that if 8, and S H _ r _ l are conjugate polar spaces, the tangent 
hyperplanes to <f> at points of the intersections of < with one of these 
are exactly the tangent hyperplanes of < which pass through the other. 

6. Prove that any plane through the vertex of a hypercone inter 
sects it in general in two straight lines, but that if n > 3, it may lie 
entirely on the hypercone. 

166. Intersection of a quadric by hyperplanes. Let < be a quadric 
hypersurface in w-space with the equation 

=- = 



It is intersected by any hyperplane If in a quadric hypersurface $ 
lying in H. To prove this we have simply to note that the equation 
of H may be taken as # w + 1 = without changing the form of (1). 

We proceed to determine the conditions under which <// is spe 
cialized. If <f) f has a singular point P, any line in If through P 
intersects < , and therefore $, in two coincident points in P. There 
fore, either H is tangent to <f> at P, or P is a singular point of <f>. 
Conversely, if H is tangent to </> at a point P, or if H passes through 
a singular point P of <, then < has a singular point at P. 

If < is a nonspecialized quadric, the hyperplane H has at most 
one point of tangency. Hence : 

/. A nonspecialized quadric is intersected by any nontangent hyper 
plane in a nonspecialized quadric of one lower dimension, and is 
intersected by a tangent hyperplane in a once-specialized quadric 
with its singular point at the point of tangency. 



POINT COORDINATES 397 

If the quadric </> is once specialized, having a singular point A, 
any hyperplane which is tangent to <j) at a point B distinct from A 
is also tangent to $ at all points of the line AB (Ex. 2, 165). 
Hence : 

II. If the quadric c/> has one singular point A, any hyperplane which 
does not pass through A intersects < in a nonspecialized quadric of one 
lower dimension; any hyperplane through A but not tangent at any 
other point intersects <f> in a once-specialized quadric, with a singular 
point at A; and any hyperplane tangent along the line AB intersects 
<t> in a twice-specialized quadric with the line AB as a singular line. 

More generally, let $> be an r-fold specialized quadric containing 
a singular <SJ_ 1? which we shall call S. Any hyperplane meets S in 
an $_ 2 or else completely contains S. Moreover, if H is tangent to 
<f> at some point P not in $, it is tangent at all points of the S r 
determined by P and , and therefore contains S. From these facts 
we have the following theorem: 

III. If the quadric $ is r-fold specialized, having a singular 
(r V)-flat S, any hyperplane H not containing S intersects <j) in an 
(r V)-fold specialized quadric whose singular (r ^-flat is the 
intersection of H and S; any hyperplane containing S but not tangent 
to </> intersects $ in an r-fold specialized quadric whose singular 
(r V)-flat is S; and any hyperplane tangent to (f) at P intersects </> in 
an (r + V)-fold specialized quadric whose singular r-flat is determined 
by P and S. 

Consider, now, the intersection of < and the two hyperplanes 



which we shall call H^ and H Z respectively. H^ intersects <f> in a 
quadric $ lying in S n _ v and H 2 intersects </> in a quadric (/>", which 
lies in the S n _ 2 formed by the intersection of H 1 and H 2 . Hence 
the common intersection of the quadric (1) and the hyperplanes (2) 
is a quadric of n 3 dimensions lying in a space of n 2 dimensions. 
This quadric is also the intersection of the quadric determined by 
<> and H l and that determined by < and H^, 

This quadric may also be obtained as the intersection of </> and 
any two hyperplanes of the pencil 

=0, (3) 



398 ^-DIMENSIONAL GEOMETBY 

in which there are in general two hyperplanes tangent to $ and 
fixing two points of tangency on </>. Hence we have the theorem : 

IV. The intersection of a quadric surface <f) by an S n _ 2 formed by 
two hyperplanes consists in general of an Sf } _ z formed by the inter 
section of two hypercones lying on cf>. The S^ 8 has the property 
that any point on it may be joined to each of two fixed points on </> 
by straight lines lying entirely on <f). 

Of course the fixed points and the straight lines mentioned do 
not in general belong to the S^ 9 . 

We shall examine this configuration more in detail for the case 
in which < is not specialized, and shall assume the equation of <f> 
in the form ^ = 0. (4) 

Then the condition that a hyperplane of the pencil (3) is 
tangent is ^+^^,+ ^^0. (5) 

If the roots of equation (5) are distinct, there are two tangent 
hyperplanes in the pencil (3), and we have the general case described 
in theorem IV. If the roots of (5) are equal, there is only one 
tangent hyperplane, and the corresponding hypercone on <f) is not 
sufficient to determine the >S 2 1 3 , but must be taken with another 
hyperplane section. 

Finally, equation (5) may be identically satisfied. This happens 

when 



which express the facts that each of the hyperplanes 11^ and JT 2 
given by equations (2) are tangent to c/>, and that the point of 
tangency of each lies on the other. Then any one of the hyper 
planes of the pencil (3) is tangent to $, and the point of tangency 
is fl f + \b { , so that the points of tangency lie on a straight line. 
The pencil of hyperplanes (3) consists, therefore, of the hyperplanes 
tangent to <f> at the points of a straight line on (f>. Let us call this 
line h. Then all points on the S^ 8 determined by <, H v and H 2 
may be joined to any point of h by means of a straight line lying 
on <. Let y. be a point on f! 3 . Then any point on the line joining 
y { to a point of h is .+ X&. + py.. The coordinates of this point 
satisfy equations (2) and (4) by virtue of (6) and the hypothesis 



POINT COORDINATES 399 

that y. satisfies these equations. Consequently in this case *S 2 1 3 is 
a specialized quadric with A as a singular line. 

Consider, now, the intersection of <f> by an S H _ S denned by the 
hyperplanes = - = 0. (7) 



These determine with $ an $,fl 4 , which may also be determined 
as the intersection of $, and any three linearly independent hyper- 
planes of the bundle denned by 

2)(a i +XJ < + ^,.)^=0. (8) 

Among these there are oo 1 tangent hyperplanes. If the equation 
of (j> is in the form (4), the tangent hyperplanes are given by values 
of X and /-t, which satisfy the equation 

5)(+^,-+/*O a =0, (9) 

and the points of tangency of these hyperplanes are then a f -f X6 t .+/*c t .. 
These points of tangency therefore form an ${ 2) , or curve of second 
order lying on <, and every point of the ^ 2 1 4 which we are con 
sidering may be joined to each point of this curve by a straight 
line on <. 

Equation (8) is identically satisfied when each of the hyper 
planes (7) is tangent to < and the points of tangency of each lies 
on the other two. Each hyperplane (8) is then a tangent hyper- 
plane, and the points of tangency are a. + \b { -f pc^ where X, ft are 
unrestricted. The bundle therefore consists of all hyperplanes 
whose points of tangency are the points of a plane lying on </>. 
Therefore each point of the /S 2 2 4 is joined to each point of this 
special plane by lines lying on $ and on the ^ 2 1 4 . Therefore the 
>S 2 1 4 is in this case a specialized quadric with that plane as a 
singular plane. 

Consider, now, the general case of the intersection of </> by the 
S r n -ic defined by the k hyperplanes 

2XX.= 0. (Z = l, 2, ...,&) (10) 

This is an S ( _ k _ v which may also be obtained as the intersection 
of (f> and any Jc hyperplanes of the system 



in which there are generally oo*~ 2 tangent hyperplanes. 



400 ^-DIMENSIONAL GEOMETBY 

In fact, if we limit ourselves to a nonspecialized </> and take its 
equation as (4), the condition that a hyperplane (11) should be 

tangentis ^W+W + >WO = 0, (12) 

and the points of tangency are then a { V+ \ap-i- - + X /fc _ 1 af 
where, of course, \ satisfy (12). These points form, therefore, a 
S { *}. 2 on </>, and any hypercone with its vertex on this S1 2 passes 
through the S^lt^ which we are discussing. We have, therefore, 
the theorem : 

V. The intersection of a nonspecialized quadric $ by an S n _ k defined 
by Jc hyperplanes is an S k _i which, in general, has the property that 
each of its points may be joined to each point of a certain Sf^_ z on < 
by straight lines lying on <$>. 

According to this theorem we have on $ two spaces, S^ k _^ and 
$i 2 -2> such that each point of either is connected to each point of 
the other by straight lines on <. It is obvious that the condition 
must hold 2 ^k^n 1. 

If n = 3, the two spaces are $ 2) and Sj?\ each of which con 
sists of a pair of points. If n = 4, the two spaces are { 2) and 
Sf\ one of which is a curve of second order and the other a pair 
of points. If n = 5, we have either an S^ connected by straight 
lines with an /S$ 2) , or an $J 2) connected in a similar manner with 
another S?\ 

In the first and last of the examples just given we have two 
spaces of the same number of dimensions occupying with respect 
to each other the special relation described in the theorem. In 
order that this should happen, it is necessary that n k \=k 2; 

7i + l 

whence k = . Hence it is only in spaces of odd dimensions 

& 

that two quadric spaces of an equal number of dimensions should 
so lie on the quadric < that each point of one is connected with each 
point of the other by straight lines on $. The number of dimen 
sions of these spaces is one less than half the number of dimensions 
of the quadric. 

Returning to equation (12) we see that it is identically satisfied 
when the hyperplanes (10) are each tangent to <f> and the point 
of tangency of each lies on each of the others. Then the system (11) 



POINT COORDINATES 401 

consists of hyperplanes tangent to (j> at the points of an S k _ l lying 
on <f). The S^^^ determined by < and (10) is then a &-fold spe 
cialized quadric with the aforementioned S k _ l as a singular locus. 
167. Linear spaces on a quadric. It is a familiar fact that straight 
lines lie on a quadric in three dimensions. We shall generalize this 
property by determining the linear spaces which lie on a quadric 
in n dimensions. Let the quadric (f> be given as in 166, and let 
S r be a linear space denned by the n + 1 equations 

pxi=^+\y?+---+\y ( r"- (i) 

The necessary and sufficient condition that z. of (1) should lie 
on <f> for all values of \ is that y { should satisfy the r+1 equations 

f^ (I = 1, 3, . . ., r +1) (2) 



and the ^ - equations 



y/vr=<>, a**), <* m=i, 2,..., 

of which the first set express the fact that each point y ( ? is on <, 
and the second set say that each point is in the tangent hyper- 
plane to (/> at each of the other points. 

Take any point P l on < and let T x be the tangent hyperplane 
at P r Then T^ intersects <f> in a specialized quadric S ( n *l 2 . Take P^ 
any point on S ( ^ 2 . The line P^ then lies on < by the conditions (2) 
and (3) and on S_ 2 , because >S 2 1 2 is specialized. The hyperplane 
T 2 tangent to <j> at P 2 is also tangent to /S^ 2 and intersects the 
latter in an S ( ^ 3 which contains P v P r T 2 will also contain other 
points of fI 3 if n 3 > 1; that is, n > 4. If this condition is met, 
take jfj in iSf! 3 but not in Pf^. The three points 7J, ^, ^ determine 
an $2 which lies on <f> by virtue of equations (2) and (3). 

The hyperplane T 3 , which is tangent to <j> at I, is also tan 
gent to S1 B and intersects it in an f! 4 which contains S^. It will 
contain other points of /S 2 2 4 if TI 4 > 2 ; that is, n > 6. If this 
condition is met we may take another point, P, on this < 2 1 4 but 
not on S! 2 . The four points ^, ^, ^J, JJ now determine an S B which 
is on <j) by the conditions (2) and (3). 

This process may be continued as long as the condition for the 
value of n found at each step is met. Suppose we have determined 



402 ^-DIMENSIONAL GEOMETRY 

in this way an $_! lying on </> by means of r points, the tangent 
hyperplanes at which have in common with (/> an S^ r _ l contain 
ing #_ r If n-f-l>r-l, that is, if r<|, this S^ r _, has 

points which are not on ^_ r Take P r + v one such point. It deter 
mines with S r _i an S r lying on </>. The process may be continued as 

long as r < - but not longer. Since the dimensions of the quadric < 

n l 
are n 1, we shall write the condition for r as r = and state 

the theorem: 

/. A nonspecialized quadric contains linear spaces of any number 
of dimensions equal to or less than half the number of dimensions of 
the quadric, but contains no linear space of greater dimensions. 

To find how many such linear spaces lie on the quadric, we notice 
that the point P l may be determined in oo 71 " 1 ways, the point P 2 in 
oo n ~ 2 ways, and so on until finally the point P r+1 is determined 
in oo n ~ r ~ 1 ways. The r-fl points may therefore be chosen in 

oo 2) ways ; but since in any S r , r + 1 points may be chosen in 

ao r(r+1) ways, the total number of S r on the quadric is oo~*~ c ~ 2) . 

The number of S r which pass through a fixed point may be 

determined by noticing that with JJ fixed, the r points P^ , j +1 

may be determined in oo 2 ways, and that in any S r the r 

points may be chosen in oo r2 ways, so that the number of different 

S r through a point is oo 2 . We sum up in the theorem : 

//. Upon a nonspecialized quadric there exist oo * 2) >$J, of 

. 7 (2n-3r-3) 

which oo pass through any fixed point on the quadric. 

If n is odd, the greatest value of r is U ~ , and there are 

2 
oo* (*-i) li near spaces of these dimensions on the quadric; if n is 

n 2 
even, the greatest value of r is , and there are oo |n(n+2) linear 

spaces of these dimensions on the quadric. 

Let us consider more in detail the case in which n is odd, and 
let us place n = 2p+l. We shall limit ourselves to a non- 
specialized quadric < and shall write its equation in the form 

<+ <+ + <+!-?- xf >+1 = 0, (4) 



POINT COOKDINATES 403 

as may be done without loss of generality. The linear space of 
the largest number of dimensions on </> is then S p , and its equations 
may be written 



2ip + lp + 1 , 5 , 

^+1 = ^+1,1*1+^ + 2,2*2 + * + a p + l,p+l X p + V 

where the coefficients satisfy the relations 



In fact, any S p is denned by p + 1 linear equations connecting the 
variables ?/. and a: f , and these equations may be put in the form (5), 
provided no one of the variables u { is missing from the equations. 
But if one of these variables is missing, it is clear that the S f p cannot 
lie on (5). The conditions (6) are found by direct substitution 
from (5) in (4). 

As a consequence of equations (6), the determinant |a rt |=l,* 
and we may divide the S p into two families, according to the value 
of this determinant. Hence we have the theorem : 

///. On a nompecialized quadric of dimensions 2p in a space of 
odd dimensions 2j9+l there are two families of linear spaces of 
dimensions p. 

Now the equations of any one S p on (4) may be written by a 
proper choice of coordinates without changing the form of (4), as 



Ui =x { . ( t = l, 2, ...,^+1) (7) 

In fact, we have simply to make a change of coordinates by 
which the right-hand members of equations (5) are taken equal to 
x\ and then to drop the primes. 

Consider, then, the intersection of (7) with any S p whose equations 
are in the form (5) with \a ik \= e, where e = 1. Then (5) is of the 

* Scott s "Theory of Determinants," p. 157. 



404 ^-DIMENSIONAL GEOMETRY 

same family as (7) when e = l, and is of the opposite family when 
e = -l. The condition for the intersection of the two S is 



a u -l 



r n * > - a 1 

p+l, 1 p + l, 2 />+l,.p+l 



(8) 



If jt? is odd, equation (8) is satisfied* always when e= l, but 
is not satisfied when e =1 unless other relations than (6) exist 
between the coefficients. If p is even, equation (8) is always satis 
fied when e = 1, but is not satisfied in general when e= l. Hence 
we have the theorem: 

IV. If p is an odd number, two linear spaces S p of opposite families 
on a quadric in a space of 2j9+l dimensions always intersect, and 
two S p of the same family do not in general intersect. If p is an even 
number, two S p of the same family always intersect, and two S p of 
opposite families do not in general intersect. 

It is easily shown that any point P on </> may be given the coor 
dinates ?^=0, z t .= 0, (i = l, 2, ...,_/?), u p + l :x p+l = l:l without 
changing the form of the equation (4). The tangent hyperplane T I 
at P l is then u p+l x p+l = 0, and its intersection with </> is the /S^_i 



Any point P 2 on this locus may be given the coordinates u { = 0, 
3,.= 0, (i = l, 2, ..., p-1*), ^:^ +1 :^:^ p+1 = l:l:l:l. The line 
is then on $. The tangent hyperplane to ^ at ^ is then 
u x-x =Q and intersects S in the S_ 



Any point -^ on this locus can now be given the coordinates u t = 0, 

and the S! 2 determined by the three points P^ P 2 , P 3 lies on </> and 
has the equations u^= = u p _ 2 = x 1 = . . . = x p _ 2 = 0, u p _ l = x p _ 1 , 

u p =^ S+i = ?>*f 

* Scott, r Theory of Determinants," p. 234. 



POINT COORDINATES 



405 



Proceeding in this way we may show that any S k (k < p~) lying 
on $ can be given the equations 

^=^=0, 

u! ~v?~; , w 

*p _ k + 1 "^p-rfc + l. 

^p+l == *>+! 

without changing the form of equation (4). 

Any S p on (/> has, as we have seen, the equations (5), and if it 
also contains all points of (9), its equations reduce to the form 

<?/ /If 1* I * . . I / W 

*-*,! ^ 

<M p _ k + 1 = x p 

qi nf 

% + 1 ^J 

where the coefficients satisfy conditions similar to (6) and 

a n ! 
^ - = e. 



a p - k, 1 * a p - k, p - k 



Without change of the form of equation (4) or (9) any one of 
these S p can be given the equations 

(12) 



In fact, we have simply to make a change of variables by which 
the right-hand members of equations (10) become x[ and then to 
drop the primes. 

The S p given by (12) will intersect any S f p given by (10) always 
in the points of S fc given by (9). In order that (12) and (10) 
should intersect in some other point not in S t , it is necessary and 
sufficient that 



a u -l 



(13) 



406 ^-DIMENSIONAL GEOMETRY 

Now if p k is an odd number, equation (13) is always satis 
fied when e = 1 ; and if p k is an even number, it is always satisfied 
when e= 1. Further, we notice that if (12) and (10) have in 
common a point P which is outside of S k , they have in common 
the S k+l determined by S f k and P\ and since (12) and (10) are on <, 
this S k+l is on <. Moreover, p k is odd if p is odd and k even 
or if p is even and k odd, and p k is even if both p and k are 
odd or if both p and k are even. 

From this we have the following results : 

1. If p is odd and two S p of the same family intersect in an S k 
where k is even, they intersect in at least an S k+l . 

2. If p is odd and two S p of opposite families intersect in an S k 
where k is odd, they intersect in at least an S k+1 . 

3. If p is even and two S p of the same family intersect in an S k 
where k is odd, they intersect in at least an S k+1 . 

4. If p is even and two S p of opposite families intersect in an S k 
where k is even, they intersect in at least an S k+l . 

This may be put into the following theorem, with reference also 
to theorem IV: 

V. If p is odd, two S^ of the same family do not in general inter 
sect, but may intersect in an S k where k is odd; and two S p of opposite 
families intersect in general in a point, but may intersect in an S k where 
k is even. If p is even, two S p of the same family intersect in general 
in a single point, but may intersect in an S k where k is even ; and two 
S p of opposite families do not in general intersect, but may intersect in 
an S k where k is odd. 

If in equations (10) we take Jc=p 1, they reduce to 

u i = a n x v u i = x 0* = 2, 3, ., p + 1) 
with a n = e = \. Hence we have the theorem : 

VI. Through any S p _^ on $ go two S p , one of each family. 

More generally the number of independent coefficients in (10) is 

known from the theory of determinants to be i*- "- - 

Hence we have the theorem: 

VII. Through any S t on $ go QO <p-*><i -*-i> ff of each family. 



POINT COORDINATES 407 

EXERCISES 

1. Show that if S r lies on <f> it must lie in its reciprocal polar space. 
From that deduce the condition r ^ 

2. Prove that there are co 2 S . on < by determining the 
number of solutions of equations (2) and (3), remembering that each of 
the r -f 1 points may be taken arbitrarily on S}.. 

3 . Show that through every /. lying on <f> there pass oo~2~ c "" 8) ^ 



which lie on the quadric ( k 



n-T 



168. Stereographic projection of a quadric in S n upon S^. Let $ 
be a quadric hypersurface of dimensions n 1 in $ u , 2 any hyper- 
plane in B , so that 2 is an S^_ v and any point 011 (/>. Straight 
lines through intersect $ and 2 in general in one point each, 
and set up, therefore, a point correspondence of <f> and 2 which 
in general is one-to-one. There are, however, on both $ and 2 
exceptional points. On c/> the point is exceptional, since lines 
through and no other point of </> lie in the tangent hyperplane 
at 0, the intersection of which, with 2, is an /S^_ 2 which we shall 
call TT. Hence corresponds to any point of TT. On 2 the points 
in which the straight lines on (j> through intersect 2 are excep 
tional, since each of these points corresponds to an entire straight 
line on $. These straight lines are the intersections of <f> (/SJ 2 ^) 
and the tangent hyperplane (^_ a ) at 0, and therefore intersect 
2 (^.0 in an S^ 2) _ B which we shall call H. Evidently H lies in TT. 

These statements, which are geometrically evident, may be verified 
by the use of coordinates. Let x l : x^:* : x n + l be coordinates of a 
point in >S n , and let + ^ . . . + = 



be the equation of <. Without loss of generality we may take 
as : : : i : 1 and the equation of 2 as x n = 0. 

The equations of a straight line through and any point P 
of are, then, 



pX tl=l = 0+\x n _ v (2) 

P X n =t + X* n , 



408 ^-DIMENSIONAL GEOMETRY 

and OP meets 2 in the point Q, obtained by placing X n = in (2). 
This determines X, and the coordinates of Q are found to be 

?i : r : f-i : : .= *i : x* "-- : x n-i- : &,+ z n+1 , 
where f f are coordinates of points in 2, and #. are coordinates of 
points on </>. Since x { satisfy equation (1) we may write the 
relation between P and its projection Q in the form 



.-,= ?.-... (3) 



Equations (3) show that to a definite point P corresponds a 
definite point Q, except that the point gives an indeterminate Q 
on the locus f B = 0, which is, therefore, the equation of TT in 2. 
Also any point Q corresponds to a definite point P, except that 
any point in the locus f n = 0, *+ f. 2 2 -f- + f j f_ 1 = gives an inde 
terminate point P, but such that P and Q lie on a straight line 
through 0. Therefore 

f.= 0, #+++ -,= (4) 

are the equations which define the quadric ft. We may note that 

any point Q which is on TT but not on ft gives the definite point 0. 

Any Si which lies on $ projects into an Sf. on 2. For the 

equations ^ = x w +x ^*>+ . . . 

become by the transformation (3) 



An /S on <^> intersects the tangent hyperplane at in an Sj._ l which 
projects into an Sl_ 1 in 2. But all points of the tangent hyper- 
plane project into points on ft, and therefore this /S^ lies entirely 
on ft. Therefore we say : 

/. By stereographic projection any linear space S k lying on a quad 
ric hypersurface c/> in a space of n dimensions is brought into corre 
spondence with a linear space Sj^_ 1 lying on a quadric surface ft in 
a space of n 2 dimensions. 

This being proved, let us consider the case in which n is an odd 
number 2 p + 1. Then < is of dimensions 2 jo, and ft is of dimensions 



POINT COORDINATES 409 

2p 2. On (f> there exist linear spaces S p which project into linear 
spaces of the same number of dimensions, which we call 2 since 
they are in 2. Any two 2^ intersect in at least a point, since they 
lie in a space of 2jt? dimensions ( 164). If that point of inter 
section is not on 12, it corresponds to an intersection of the two S p on 
<, since outside of H any point of 2 corresponds to a definite point 
of <. If, however, the intersection of two 2^ lies on H, the two 
corresponding S p on < do not in general intersect. In fact, the in 
tersection of two 2_^ on II simply means that a straight line from 
in the tangent hyperplane at meets each of the two correspond 
ing S p . Since we are talking of two S p in general, their intersection 
in the tangent hyperplane at O may be considered as exceptional, 
so that we have the theorem : 

//. If two S p on the quadric <f> intersect, the corresponding S p _^ on 
the quadric fl do not in general intersect ; and if two S p on <j> do not 
intersect, their corresponding Sp_ l on H in general intersect in a point. 

In a similar manner the question of the intersections of linear 
spaces S p _ 1 on an S_ 2 may be reduced to the question of the inter 
section of two S p _ z on an S p _^ and eventually to the intersection 
of two S[ on an /S 2) ; that is, of two straight lines on a quadric sur 
face in ordinary three space. 

We may, accordingly, divide the S p on < into two families, accord 
ing as they correspond by this successive projection to the two 
families of generators on an ordinary quadric surface. From 
theorem II, however, it is evident that we have the same classi 
fication as that made algebraically in 167; for it follows that 
two S p of the same family do or do not intersect according as p is 
even or odd, and two S p of opposite families do or do not intersect 
according as p is odd or even. Exceptions may, of course, occur, 
as has been shown in 167. 

Let us consider now the intersection of </> by any hyperplane 

a i x i+ a &+ + a n x n+ + A + i= > 

which passes or does not pass through the center of projection 0, 
according as ^,~l~ rt n+i is or is not 0. The intersection with </> is 
an S1 2 which projects upon 2 into a 2jf! 2 , with the equation 



410 JV-DIMEKSIONAL GEOMETRY 

This is in general a 2^ which contains H, but if ia n + a n+1 = 0, 
it splits up into the hyperplane TT and a general hyperplane 7. 
Hence the theorem: 

If an S1 2 upon c/> does not pass through 0, it projects into a quadric 
in 2 which contains H; if an S^ 2 on $ does pass through 0, it projects 
into a hyperplane in 2 together with the hyperplane TT. 

EXERCISES 

1. Show that any tfjj, on < not passing through projects into a 2^ 
in 2 which intersects TT in a 5jJ,_i contained in O. 

2. Show that any 54 _ x not passing through intersects < in an S n 2 l 2 
which projects into a 5JL 2 which passes i times through O. 

169. Application to line geometry. Since line coordinates con 
sist of six homogeneous variables connected by a quadratic rela 
tion, a straight line in ordinary space may be considered as a point 
on a quadric surface in an S & . We shall proceed to interpret in line 
geometry some of the general results we have obtained. In so 
doing we shall, to avoid confusion, designate a point, line, and plane 
in $ 5 by the symbols /SJ, S(, S! 2 , respectively, reserving the words 
"point," "line," and "plane" for the proper configurations in S 3 . 
Let <f> be the quadric whose equation is the fundamental relation 
connecting the coordinates of a straight line. Then an S r on </> is a 
straight line, an S[ on (f> is a pencil of straight lines, and an S 2 on <f> 
is either a bundle of lines or a plane of lines. These statements are 
established by comparing the analytical conditions for pencils and 
bundles of lines given in 131 with those for S( and S 2 on </>. 

The two families of S! 2 on <f> are easily distinguished, the one 
consisting of lines through a point, the other of lines in a plane. 
It is evident that two Si of the same family intersect in an S r Q , for 
two bundles of lines or two planes of lines have always one line in 
common. On the other hand, a bundle of lines and a plane of 
lines do not in general have a line in common ; that is, two S! 2 of 
different families do not in general intersect. If, however, a point 
of lines and a plane of lines have one line in common, they will 
have a pencil in common ; that is, if two Si of different families 
on <f> intersect in an /SJ, they intersect in an S[. This is in accord 
with theorem V, 167. 



POINT COORDINATES 411 

A linear line complex is an Sf } formed by the intersection of </> 
and an S 4 . If the S 4 is tangent to <, the complex is special and con 
sists of oo 2 $( joining the points of the complex to a fixed S . The 
special linear complex in line geometry consists, therefore, of oo 2 
pencils of lines containing a fixed line. 

A linear line congruence consists of an S^ formed by the inter 
section of <f> and two iSJ. Therefore it consists in general of lines 
each of which belongs to two pencils containing, respectively, one of 
two fixed lines. When the two fixed lines intersect, the congru 
ence splits up into a bundle of lines and a plane of lines, w r ith a 
pencil in common. That suggests the theorem that on $, if the 
two fixed SQ connected with a congruence S^ lie on an S[ of <, the 
S ( ^ splits up into two S z of different families intersecting in this S[. 

A linear series is an $J 2) determined by the intersection of <J> and 
three /SJ. From the general theory we see that the series consists 
of oo 1 lines, each of which lies in a pencil containing each of oo 1 
fixed lines. It therefore consists in general of oo 1 lines intersecting 
another oo 1 lines. We leave to the student the task of considering 
the special cases of a line series. 

A linear complex 



+ a H + l X n + l = (1) 

is fully determined by the ratios a l : a 2 : : n+1 , which may be 
taken as the coordinates of the complex, and we may have a 
geometry in which the line complex is the element. 

The quantities a x : 2 : : a n + 1 are also the coordinates of a point 
in $ 5 , which is the pole of the hyperplane (1). Therefore the 
point a i is not on the quadric </> unless the complex is special. An 
S Q in S 6 is therefore a line complex. The lines of the complex a { 
correspond to the points in which the polar (1) of the point a. 
intersects <. If S^ is on c>, the complex is special and may be 
replaced by its axis so as not to contradict the previous statement 
that an 8[ on < is a straight line. In fact, if the equation of $ is 
taken as ^2^ = 0, the coordinates of a special complex and of its 
axis are the same. 

Consider now two complexes a. and b { as two points S Q in S 6 . 
They are said to be in involution if each /SJ lies on the polar plane of 
the other. From this it follows at once that if one of the complexes 



412 ^-DIMENSIONAL GEOMETRY 

is special, its axis is a line of the other ; so that if both are special, 
their axes intersect, and conversely. In case neither complex is 
special, the S denned by a { and b { are not lines in S B , and we must 
look for other geometric properties of complexes in involution. 

In S 6 the coordinates a i and b { have a dualistic significance. On 
the one hand, they are coordinates of two /SJ ; on the other hand, 
they are coordinates of two hyperplanes, the polars of these points. 
The two S f determine a pencil of S Q which lie in an S[, and the two 
hyperplanes a pencil of hyperplanes which have an $J in common. 
The pencil of S f Q contains two S Q on </>, and the pencil of hyperplanes 
contains two hyperplanes tangent to (f>. It is then evident that 
two complexes are in involution when the two S Q in S b which represent 
them are harmonic conjugates with respect to the quadric <, or, what 
is the same thing, when the two hyperplanes defining the com 
plexes are harmonic conjugates to the two tangent hyperplanes to 
<f> which are contained in the pencil defined by the two complexes. 

It is clear that in any pencil of complexes the relation between 
a complex and its involutory complex is one-to-one. 

If we consider a fixed complex a f , all complexes in involution to 
it are represented by points in an $[, which is the polar hyperplane 
of a. with respect to </>. 

This relation can be generalized. Let S k be a linear space of 
points in $ 6 , and let ^_ t be the conjugate polar space with respect 
to </>, so that any point in S f k is the harmonic conjugate with respect 
to </> of any point in S^_ k . We have, then, two series of complexes, 
each of which is in involution with each one of the other series. 
The points in which S k intersect </> are special complexes. Their 
axes, therefore, must lie in each of the complexes in Sl_ k j as has 
been shown above. In other words, the axes of the special complexes 
of one series are the straight lines common to the complexes of the 
involutory series, and conversely. The proof of the converse is left 
to the student. 

For example, consider the pencil of complexes a i + \b. in invo 
lution with the series of complexes c^X d^^ e^v f^ The pencil 
of complexes have in general a congruence of straight lines in 
common, and these are the axes of the special complexes of the 
series. On the other hand, the series of complexes have in general 
two lines in common which are the axes of the special complexes 



POINT COORDINATES 413 



of the pencil. Again, consider the bundles of complexes a i 
and { +A//-h /* #; in involution. The complexes of either bundle 
have in common the oo 1 straight lines of a regulus which are the 
axes of the special complexes of the other bundle. 

Any collineation of S 6 is a transformation of S s by which a 
linear line complex goes into a linear line complex, and any linear 
series of complexes goes into another such series. If, in addition, 
the quadric $ is transformed into itself, straight lines in S 8 are 
transformed into straight lines, and any $J on < is transformed 
into another S! 2 on <. But as there are two systems of S f 2 on $, 
the transformation may transform an S 2 either into one of the same 
system or into one of the other system. In the first case, points 
in $3 are transformed into points ; in the second case, points in $ 3 
are transformed into planes. We have, accordingly, the theorem: 

A collineation in $ 5 which leaves the quadric <f> unaltered is either 
a collineation or a correlation in S 3 . 

EXERCISES 

1. Discuss oriented circles in a plane as points on a quadric in S 4 . 

2. Discuss oriented spheres in ordinary space as points on a quadric 
in S 6 . 

170. Metrical space of n dimensions. We have been considering 
spaces in which a point is defined by the ratios of homogeneous 
variables. We may, however, consider equally well a space in 
which the point is defined directly by n coordinates u^ u 2 , , u n , 
and where the equations are not homogeneous. All equations may 
be made homogeneous, however, by placing 



The discussion is then reduced to the homogeneous case, but 
the use of t as the n + \st coordinate emphasizes the unique char 
acter of that coordinate. In fact, when t 0, some or all of the 
original coordinates become infinite. This enables us to handle 
infinite values of the original coordinates. Such sets of values 
may be distinguished from each other by the ratios of #,, so that 



414 ^-DIMENSIONAL GEOMETRY 

is said to define a definite point at infinity. We have, therefore, 
a special case of protective space with a unique hyperplane t 0. 
We may define a distance in a manner analogous to that used 
in three dimensions, by the equation 

d* = (u[ - uy + <X - ,) 2 ++- nj\ (2) 

or, in homogeneous form, 






From this it appears that the distance between two points can be 
infinite only if t or t r is zero. Conversely, with the exception noted 
below, a point for which t = is at an infinite distance from any 
point for which t ^ 0. Therefore t = is called the hyperplane at 
infinity. 

On the hyperplane at infinity the coordinates are projective 
coordinates in ^ n-1 defined by the ratios x^: x 2 : : x n . 

An exception to the statement that points on the hyperplane at 
infinity are at an infinite distance from points not on that hyper 
plane occurs for points on the locus 

*=0, ^ 2 +z 2 2 +... + z n 2 =0, (4) 

since the distance of any point on this locus from any other point 
is indeterminate. This locus, which is an S*_ 2 , or a quadric hyper- 
surface in the hyperplane at infinity, is called the absolute. 

The following properties of metrical space are such obvious 
generalizations of those of three-dimensional space that a mere 
statement of them is sufficient. 

A hypersphere is the locus of points equidistant from a fixed 
point. Its equation is 



and it is obvious that all hyperspheres contain the absolute, but 
no other point at infinity. 

A straight line may be defined by the equations 

X 1 -a l _x 2 -a 2 _ _x n -a n 

~ ~~ 



This line meets the hyperplane at infinity in the point ^: ? 2 : : l n . 
Hence, through any point in space go oo 71 " 1 lines distinguished by 



POINT COORDINATES 415 

the ratios of the quantities l t . We say that these quantities deter 
mine the direction of the line, direction being that property which 
distinguishes between straight lines through the same point. 

Two lines with the same direction meet the hyperplane at 
infinity in the same point and are called parallel. Two lines with 
directions 1. and V. meet the hyperplane at infinity in two points 
with coordinates l t and ??, and the straight line connecting these 
two points meets the absolute in two points such that the cross 
ratio of the four points is 



We shall define this as the cosine of the angle between the two 
lines; namely, 



In particular two lines are perpendicular when 

yi+yi+--- + yi=o. 

A line meets the absolute when, and only when, 

?+?+ + y=- 

In that case the distance between any two points on the line is 

zero, and the line is a minimum line. Through any point of space 

go, then, oo n ~ 2 minimum lines forming a hypercone of oo"" 1 points. 

A tangent hyperplane to a hypersphere intersects it in oo n ~ 8 lines, 

and since the sphere contains the absolute these are minimum lines. 

Any hyperplane 

!#!+ a 2 x 2 -\ ---- + <* n x n + a n+l t = 

meets t = in the locus 

!#! + flfyEgH ---- -f a n x n = 0, 

which is a hyperplane in the S n _^ defined by t = 0. It is tangent 
to the absolute when 2j| f=a 0. 

Hyperplanes satisfying this condition are minimum hyperplanes ; 
all others are ordinary hyperplanes. 

The intersection of an ordinary hyperplane with t = has a pole 
with respect to the absolute whose coordinates are a x : a. 2 : : a B , 



416 ^-DIMENSIONAL GEOMETRY 

and any straight line with the direction a 1 : a 2 : - : a n is said to be 
perpendicular to the hyperplane. In fact, from the definition of 
perpendicular lines already given, this line is perpendicular to any 
line in the hyperplane, and conversely. 

Two hyperplanes are perpendicular when the pole of the trace 
at infinity of either contains the pole of the trace of the other. 
Therefore the condition for two perpendicular hyperplanes is 

A +&+ +A=o. 

It follows that the n hyperplanes 



are mutually perpendicular hyperplanes intersecting at 0. Through 
or any point of space pass an infinite number of such mutually 
orthogonal hyperplanes ; for, as seen in 165, we may find in t = 
an infinite number of coordinate systems such that the absolute 
retains the form ^xf = 0, and the lines drawn from to the points 
#;= 0, z k = Q (Jc= i) determine the hyperplanes required. 

In this way any ordinary hyperplane may be made the plane 
x n = 0. The coordinates in this hyperplane are x l : x 2 : - - : x n _ l : t, 
and its absolute is t = 0, x? + x* + - . + x*^ = 0. 

Therefore the geometry in any ordinary hyperplane differs from 
that in the original space only in the number of the dimensions. 
Two linear spaces, S^ and S^ are said to be completely parallel 
if they intersect only at infinity and if the section of S r at infinity 
is completely contained in the section of /SJ at infinity (r^r^). 
Since the section of /S^ at infinity is an S r _ 15 it is necessary that 
S^ and S rz should lie in an $ t + rf _ (Pi _ 1) = S r ^ (theorem IV, 164). 
Moreover, if we take r^ points in the S ^ at infinity, one other 
point not at infinity in S r , and r z r^l points not at infinity in 
$ 2 , we have r 2 + 2 points to determine an $ +r Therefore, 

If two linear spaces S^ and ^( r i^ r O are completely parallel, they 
lie in an S r ^ +l and completely determine it. 

Consider now two spaces, ^ and $ (r ^ r r ), which do not in 
tersect (T-J+ r 2 < n). They determine in the hyperplane at infinity 
two nonintersecting spaces, S^ and $ r If we take ^ points in 
^ 4 -ii an( i r 2 points in S r ^_ v we determine, by means of these points, 
an ^i+ - 2 - 1 in tlie ^yperplaue at infinity which contains both S f r _ 1 



POINT COORDINATES 417 

and S r<t _ r By means of this , +r _ 1 and one other point in S r not 
at infinity, we determine an fij! +r which contains S r because it 
contains ^ + 1 of its points, and is parallel to S r since the inter 
section with infinity of S r r is completely contained in that of S r +r . 
Hence, 

If S f r and /SJ are two nonintersecting linear spaces with r l = r 2 , it 
is possible to pass a linear space S, + through S r parallel to S . / 

It is obviously possible to define as partially parallel two linear 
spaces which intersect at infinity and nowhere else. This would 
lead to a series of theorems of which those in 158 are examples, 
but we shall not pursue this line of investigation. 

Two linear spaces will be defined as completely perpendicular 
when each straight line in one is perpendicular to each straight 
line of the other. If $ and <SJ are two linear spaces intersecting 
the hyperplane at infinity in S . _ 1 and S f r _ v respectively, it follows 
that the necessary and sufficient condition that S r should be com 
pletely perpendicular to /S^ is that _ l should lie in the conjugate 
polar space of S,^ with respect to the absolute, when, of course, 
$ 2 _i will also lie in the conjugate polar space of S r _ x with respect 
to the absolute. 

Now the conjugate polar space of S f r in S n _ l (the hyperplane at 
infinity) is, by 165, /^.^.j. If ^ is given, its intercept on the 
plane at infinity <SJ j is determined, and the reciprocal polar space 
$, _ ri _i is also uniquely determined. One other point in finite 
space then determines with this S n _ .^ an S^_^ which is completely 
perpendicular to the given S .. Hence the theorem. 

Through any point in space one and only one S n _ r can be passed 
which is completely perpendicular to a given S r . Any linear space 
contained in S n _ r is then completely perpendicular to any linear space 
in S f r . 

It is possible to define as partially perpendicular, spaces each 
of which contains a straight line perpendicular to the other, as in 
166, but we shall not do this. 

Let us consider the stereographic projection of a hypersphere upon 
a hyperplane. Here we have merely to use the results of 168, 
interpreting the quadric (f> as a hypersphere, and the plane # n+1 = 



418 ^-DIMENSIONAL GEOMETRY 

as the hyperplane at infinity in S n . Then TT is the hyperplane at 
infinity in S n _ v and n is the absolute. We have at once the theorem : 

By the stereographic projection of a hypersphere in S n upon a 
hyperplane S n _ v hyperplanar sections of $ go into hyperplanes or 
hyper spheres of S n _ l according as the hyperplanar sections of <f> do or 
do not contain the center of projection. 

A collineation in S n by which </> is invariant gives a point trans 
formation on (f> by which hyperplanar sections go into hyperplanes. 
There is a corresponding transformation in S n _ l by which a hyper 
plane or a hypersphere goes into either a hyperplane or a hypersphere. 
If the collineation in S n leaves as well as <j> invariant, hyper 
planes of S n _ l are transformed into hyperplanes, and the transfor 
mation is a collineation. But the transformation in S n leaves the 
tangent hyperplane at unchanged, and therefore the correspond 
ing transformation in S n _ l leaves the absolute unchanged. Hence, 

Collineations in S n which leave <j) and the point on <f> unchanged 
determine collineations in S n _ l which leave the absolute unchanged and 
which are therefore metrical transformations. 

Collineations in S n which leave </> but not unchanged determine 
point transformations in S n _ l by which hyperspheres go into hyper- 
spheres, a hyperplane being considered a special case of a hypersphere. 

We have used in 168 one set of coordinates (#,) for the points 
of <, and another set (f.) for the points of S n _ v but clearly the 
coordinates x { may also be used to determine points in S n _ r 

We shall have, then, for the points of S n _ l n + l homogeneous 
coordinates connected by a quadratic relation, and such that a 
linear equation between them represents a hypersphere with the 
hyperplane as a special case. Each of the coordinates x i equated 
to zero represents a hypersphere. We may, accordingly, call them 
(n -j-1) -poly spherical coordinates of the points of S H _ r They are a 
generalization of the pentaspherical coordinates of S & . We say: 

Protective coordinates of points on a hypersphere in S n are poly- 
spherical coordinates of points on an S n _ l into which the hypersphere 
is stereographically projected. Collineations of 8 H which leave the 
hypersphere invariant are linear transformations of the polyspherical 
coordinates o $_ 



POINT COORDINATES 419 

171. Minimum projection of S n upon S n _ 1 . Consider in S H , with 
nonhomogeneous metrical coordinates, the minimum hypercone 

(*.- >) 2 + (*,- 2 ) 2 + -.+(*- <O 2 = o. (i) 

The section of this by the hyperplane x n = is 

(*,- !) +(*,-,) + +.CUl- n-.) 2 + S = 0. (2) 

which is a hypersphere in the fl n _ l defined by x n = 0. We say that 
the vertex a i of the minimum hypercone (1) in S n is projected min 
imally into the hypersphere (2) in S n _ lf Obviously, in order that 
the hypersphere (2) should be real the vertex of (1) must be imag 
inary. More exactly the coefficients 1? 2 , , a n _ l must be real and 
a n pure imaginary. 

The coordinates of the vertex of a hypercone in S n are then 
essentially elementary coordinates ( 146) of a hypersphere in S n _ v 
but the radius of the sphere is ia n instead of a n . Let us, however, 
introduce into S n polyspherical coordinates based upon n + 2 hyper- 
spheres. The coordinates of the vertex of a hypercone in S n and, 
consequently, of a hypersphere in /S r n _ 1 are then n + 2 homogeneous 
coordinates connected by a quadratic relation. They are therefore 
higher sphere coordinates of oriented hyperspheres in >S n _ l . But 
we have seen that the polyspherical coordinates in S n are projec- 
tive coordinates of points on a hypersphere in S H+l . We have, 
therefore : 

The protective coordinates of a point on a hypersphere in S n+l 
become, by stereographic projection, the n + 2 polyspherical coordinates 
of a point in S n , and, by further minimum projection, the higher sphere 
coordinates of a hypersphere in S n _ r 

We have in this way obtained a geometric construction by which, 
for example, oriented spheres in S 3 may be brought into a one- 
to-one relation with points on a hypersphere in S 6 . 

EXERCISES 

1. Show analytically that a point x 1 : x. 2 : >:x n+1 on the hyper 
sphere x} + xl 4- + a-J + 1 = in S n projects by the double projection 

of the text into the hypersphere (ix n + x n + l ) (? -\ h J_ 2 ) - 2 a^, 

2z__+w?-x= in 2_. 



420 



^-DIMENSIONAL GEOMETRY 



2. Establish the following relations between 5 fi , S 4 , and S 8 , <f> being 
a hypersphere in S : . 



A point on <. 

A hyperplane sec 
tion of <. 

A section of <f> by a 
tangent hyperplane. 

A minimum line 
on <. 

A minimum plane 
on <. 

A section of <j> by 
any SJ. 

A minimum curve 
on <. 



A point. 
A sphere. 

A point sphere. 
A minimum line. 

A minimum plane 
of second kind. 

A hypersurface of 
order y. 

A minimum curve. 



A sphere. 

A sphere complex. 

A special sphere 
complex. 

A pencil of tangent 
spheres. 

A bundle of tan 
gent spheres. 

A sphere complex 
of order g. 

A series of oo 1 
spheres, each of which 
is tangent to the con 
secutive one. 



REFERENCES 

Sphere geometry : 

COOLIDGE, Line and Sphere Geometry (see reference at end of Part III). 
Line geometry : 

HUDSON, Rummer s Quadric Surface. Cambridge University Press. 

JESSOP, Treatise on the Line Complex. Cambridge University Press. 

KCENIGS, La ge ome trie re gle e et ses applications. Gauthier-Villars. 

FLU CUE R, Neue Geometric des Raumes. Teubner. 

Pliicker s work is the original authority. It is quoted here for its historical 
value. The student will probably find it more convenient to consult the other 
texts, the scope of which is sufficiently indicated by their titles. 

Geometry of n dimensions : 

JOUFFRET, Ge ome trie a quatre dimensions. Gauthier-Villars. 

V-MANNING, Geometry of Four Dimensions. The Macmillan Company. 

< 
Manning s book is synthetic/ Jouffret s analytic. Especial mention should be 

made of the historical account in Manning s introduction, with copious references 
to the literature. 

For general n-dimensional geometry reference will be made to the following 
journal articles, which the author has found especially useful in preparing his text : 

KLEIN, ff Ueber Liniengeometrie und metrische Geometric." Mathematische 

Annalen, Vol. V, 1872. 
SEGRE, " Studio sulle quadriche in uno spazio lineare ad un numero qualunque 

di dimension!. " Memorie della reale accadernia delle scienze di Torino. 

Second Series, Vol. XXXVI, 1885. 
VERONESE, " Behandlung der projectivischen Verhaltnisse der Raiime von ver- 

schiedenen Dimensionen durch das Princip des Projicirens und Schneidens." 

Mathematische Annalen, Vol. XIX, 1882. 



INDEX 



(The numbers refer to pages) 



Absolute, 370 

Affine transformation, 102 

Angle, 105, 107, 188, 254, 369, 415; 
between circles, 144 ; between spheres, 
286, 344 ; of parallelism, 112 

Asymptotes, 33 

Asymptotic lines, 361 

Axis, of range, 8 ; of pencil of planes, 
12; of quadric, 230; of special line 
complex, 311 ; of any complex, 318 

Base of range, 8 

Bicircular curve, 174, 281 

Brianchon s theorem, 76 

Bundle, of planes, 196, 198 ; of spheres, 
268, 293, 342; of lines, 306; of tan 
gent spheres, 351 

Center, of conic, 32 ; of quadric, 224, 227 

Characteristic of surface, 211 

Circle, 30; at infinity, 181 

Circle coordinates, 171, 177 

Circle points at infinity, 30, 105 

Clairaut s equation, 137 

Class, of curve, 55 ; of surface, 207 ; of 
line complex and congruence, 308 

Clifford parallels, 255 

Collineations, 72, 240, 250, 413 

Complex, of circles, 149, 172, 173, 179 ; of 
spheres, 269,293,341,346, 353 ; of lines, 
308, 310, 316, 317, 328: cosingular, 
332 ; tetrahedral, 333 ; tangent, 353 

Conformal transformation, 126 

Congruence, of circles, 172, 179; of lines, 
308, 322, 335, 336; normal line, 338; 
of spheres, 348 

Conies, 32 ; pairs of, 95 

Conjugate points and spaces in n dimen 
sions, 393 

Conjugate polar lines, of a line complex, 
314 ; of a quadric, 223 



Conjugates, harmonic, 18 
Contact transformations, 120, 258 
Coordinates, 1, 3 ; point, 8, 27, 34, 138, 

164, 180, 193, 282, 288, 362, 388 ; line, 

11, 38, 301, 305, 410; plane, 12, 197; 

circle, 171, 177; sphere, 341, 343 
Correlation, 88, 246 
Cosines, direction, 191, 377 
Cross ratio, 16 
Curvature, lines of, 338 
Curve, 50, 58, 200 
Cuspidal edge, 212 
Cyclic, 174 
Cyclide, Dupin s, 274, 350 ; general, 279, 

297 
Cylindroid, 323 

Deferent, 299 

Degree of space in n dimensions, 390 

Desargues, theorem of, 45 

Developable surface, 208, 214 

Diameter; of parabola, 67 ; of line com 
plex, 318 

Diameters, conjugate, of conic, 64; con 
jugate, of quadric, 224 

Diametral plane, 224, 230 

Dilation, 136 

Direction, 188 ; in four dimensions, 368 ; 
in n dimensions, 415 

Distance, 29, 139, 283, 290 ; projective, 
108, 111, 115, 117, 254, 255; in four 
dimensions, 368 ; in n dimensions, 
414 

Double circle of complex, 174 

Double pair of correlation, 90, 247 

Duality, 2 ; point and line in plane, 40, 
56; tetracyclical plane and quadric 
surface, 161, 163, 250; point and 
plane, 199 ; line and sphere, 357 ; line 
in three dimensions and point in four, 
384, 410 



421 



422 



INDEX 



Ellipsoid, 228 

Elliptic space, 115, 255 

Focal curve, 300 

Focal points of line congruence, 336, 338 

Foci of conic, 65 

Form, algebraic, 140 

Generators of quadric, 232, 326 
Groups, 6 

Hexaspherical coordinates, 386 
Homology, 85 ; axis of, 49, 76 ; center 

of, 49, 76 
Hori cycle, 114 
Hyperbolic space, 110, 254 
Hyperboloid, 228 
Hyperplane, 362, 369; at infinity, 368, 

414 ; polar, 383 
Hypersphere, 370, 385\ 413 
Hypersurface of second order, 382, 392 
Hypocycle, 114 

Imaginary element, 2 

Imaginary line, 184, 191 

Imaginary plane, 187 

Infinity, 3 ; locus at, 8, 28, 139, 142, 165, 

186, 284, 368, 414 
Invariant, 7 

Inversion, 121, 124, 156, 261, 270, 291 
Involution, 15; of line complexes, 327, 

411 ; of sphere complexes, 347 

Klein coordinates, 306 
Rummer s surface, 332 



Line, equations of, 27, 35, 195, 197, 362, 
388 ; at infinity, 28 ; proper and im 
proper, 183; completely and incom 
pletely imaginary, 191 

Line coordinates, 10, 38, 301 

Line element, 133 

Lobachevskian geometry, 112 

Magnification, 104 
Minimum curves, 192 
Minimum hyperplanes, 378 
Minimum lines, 184* 189, 378 
Minimum planes, 188, 190, 285, 378 



w-line and n-point, 44 
Non-Euclidean geometry, 112 
Null sphere, 182 
Null system, 248, 321 

Order, of plane curve, 53; of surface, 
205, 220 ; of line complex, congruence, 
and series, 308 
Orientation of spheres, 344 
Orthogonal circles, 145, 172, 178 
Orthogonal spheres, 267, 286, 341 
Osculating plane, 202 

Pappus, theorem of, 48 

Parabolic space, 117, 255 

Paraboloid, 229 

Parallelism, 28, 112, 187, 370, 416; com 
plete and simple, 371 ; Clifford, 255 

Pascal s theorem, 75 

Pedal transformation, 131 

Pencil, of points, 8; of lines, 11, 37, 39, 
306; of planes, 12, 196; of conies, 64; 
of circles, 146 ; of spheres, 266, 293, 
342 ; of tangent spheres, 350 

Perpendicularity, 111, 190, 416; com 
plete perpendicularity and semiper- 
pendicularity, 375 

Perspectivity, 21 

Plane, 185, 197, 285, 362; at infinity, 
186; of lines, 307 

Plane coordinates, 197 

Plane element, 259 

Planes, completely and simply parallel, 
371; completely perpendicular and 
semiperpendicular, 375 

Pliicker coordinates, 301 

Pliicker s complex surface, 334 

Point, equation of, 39, 197 

Point-curve transformation, 127, 263, 361 

Point-point transformation, 120, 260 

Point sphere, 182, 185, 285 

Point-surface transformation, 262 

Polar, with respect to point pair, 20; 
with respect to curve of second order, 
59; with respect to curve of second 
class, 70; in general, 140; with re 
spect to surface of second order, 222 ; 
with respect to surface of second 
class, 238 ; with respect to linear line 



INDEX 



423 



complex, 315; with respect to quad 
ratic line complex, 328 ; with respect 
to hypersurface, 383, 393 

Polar lines, conjugate, 223, 314 

Polar spaces, conjugate, 393 

Power of point, with respect to circle, 
150; with respect to sphere, 287 

Projection, 20; stereographic, 162, 407, 
418 ; minimum, 419 

Projective geometry, in plane, 101 ; in 
three dimensions, 249; on quadric sur 
face, 250 ; in n dimensions, 388 

Projective measurement, 107, 253 

Projectivity, 13, 20 

Pseudo circle, 113 

Quadrangle, complete, 44 
Quadrilateral, complete, 44 

r-flat, 388 
Radical axis, 268 
Radical center, 269 
Radical plane, 267 
Range, of points, 8 ; of conies, 71 
Ratio, anharmonic, 17; cross, 16; har 
monic, 18 
Reflection, 104 
Regulus, 326 
Relativity, 119 
Riemannian geometry, 116 
Ring surface, 275 
Rotation, 103 
Rulings on quadric, 232 

Series, line, 308, 324 ; sphere, 349 
Sheaf of planes, 12 
Singular complex of circles, 174 
Singular lines, 54, 67, 329 



Singular planes, 216, 222, 329 
Singular points, 52, 58, 206, 296, 329, 383 
Space, linear, 388 ; on quadric, 401 
Specialized quadric, 395 
Sphere, 266, 284 ; oriented, 344 
Sphere coordinates, 341 
Spherical geometry, 116 
Spheroquadric, 281 

Surface, in point coordinates, 205; in 
plane coordinates, 215 ; anallagmatic, 
274, 299; singular, 331, 355; Rum 
mer s, 332 ; Pliicker s, 334 

Tangent circles, 178, 295 

Tangent hyperplanes, 383 

Tangent line, to curve, 51, 200; to sur 
face, 205 

Tangent line complexes, 328 

Tangent plane to surface, 206 

Tangent planes, 345 

Tangent sphere complexes, 353 

Tangent spheres, 295, 345, 350 

Tetracyclical coordinates, 138 

Thread, 25, 142 

Transform of an operation, 5 

Transformation, defined, 4 ; affine, 102 ; 
contact, 120, 258 ; inversion, 124, 156, 
261, 270, 291; linear, 13, 78, 88, 154, 
169, 240, 246, 291; metrical, 101, 155, 
249, 291 ; projective, 20, 100, 249, 253 ; 
pedal, 131; point-point, 120, 260; 
point-curve, 127, 263 ; point-surface, 
262 ; quadric inversion, 121 ; recipro 
cal radius, 124, 261, 270 

Translation, 103 

Union of line elements, 134; of plane 
elements, 260 



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